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Robust Mean-Covariance Solutions for Stochastic Optimization
- Magnus Ray
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1 OPERATIONS RESEARCH Vol. 00, No. 0, Xxxxx 0000, pp issn X eissn INFORMS doi /xxxx c 0000 INFORMS Robust Mean-Covariance Solutions for Stochastic Optimization Ioana Popescu INSEAD, Decision Sciences Area, Blvd. de Constance, Fontainebleau, France. We provide a method for deriving robust solutions to certain stochastic optimization problems, based on mean-covariance information about the distributions underlying the uncertain vector of returns. We prove that for a general class of objective functions, the robust solutions amount to solving a certain deterministic parametric quadratic program. We first prove a general projection property for multivariate distributions with given means and covariances, which reduces our problem to optimizing a univariate mean-variance robust objective. This allows us to use known univariate results in the multidimensional setting, and to add new results in this direction. In particular, we characterize a general class of objective functions (so called one or two-point support functions), for which the robust objective is reduced to a deterministic optimization problem in one variable. Finally, we adapt a result from Geoffrion (1967a) to reduce the main problem to a parametric quadratic program. In particular, our results are true for increasing concave utilities with convex or concave-convex derivatives. Closed form solutions are obtained for special discontinuous criteria, motivated by bonus and commission based incentive schemes for portfolio management. We also investigate a multi-product pricing application, which motivates extensions of our results for the case of non-negative and decision dependent returns. Subject classifications : programming: stochastic, quadratic; decision analysis: risk; finance: portfolio. Area of review : Decision Analysis. History : Received January 2004; accepted December
2 2 Operations Research 00(0), pp , c 0000 INFORMS 1. Introduction In this paper we provide a general approach for deriving robust solutions for stochastic optimization programs, based on mean-covariance information about the distributions underlying the return vector. Consider a decision maker interested in maximizing his expected utility u of a linear reward function, subject to a given set of convex constraints P on the vector of decision variables x. The decision maker solves the following general stochastic mathematical program without recourse: (P ) maximize E[u(x R)] subject to x P, Throughout the paper, random variables and random vectors will be denoted in bold, to be distinguished from deterministic quantities. We will use max and min in the wide sense of sup and inf, to account for the cases when the optimum is not achieved. When the distribution of the uncertain return vector R is exactly known, Problem (P) is a general one stage stochastic program. One obvious case when the solution of Problem (P) does not depend on the actual distribution of R, except its mean and covariance, is for quadratic utilities u(r) = ar 2 + br + c, in which case it amounts to: max x P ax (Σ + µµ )x + bx µ + c. However, this is only the case for quadratic (or linear) utility functions, explaining in part why these classes are particularly favored in economics and financial models. In many applications of stochastic programming, there is considerable uncertainty regarding the distribution of R. In many instances, only partial information about this distribution is available, or reliable, such as means, variances and covariances. A common approach in the literature is to assume that the underlying distribution is multivariate normal, and solve the corresponding stochastic program. However, this assumption is hard to justify unless the randomness comes from a central limit theorem type of process. Alternatively, the robust approach consists in basing the decision strictly on the available information, and maximizing the worst case expected utility over all possible distributions with the given properties. Even if the underlying distribution is actually known, the robust approach can be used to provide tractable approximations for the solution of Problem (P), which is typically a very difficult problem. The robust problem is the focus of our investigation, and can be formulated as follows: (G) max x P min R (µ,σ) E[u(x R)], (1) where we denote R (µ, Σ) to represent the fact that the distribution of R belongs in the class M n (µ,σ) of n-variate distributions with mean µ and covariance Σ. By a simple change of variable
3 Operations Research 00(0), pp , c 0000 INFORMS 3 u = u we obtain the min-max version of Problem (G). All the results in this paper can be rephrased to hold for min-max problems, so for simplicity we restrict the analysis to the max-min case. As an example, consider a portfolio manager interested in choosing a portfolio x among n assets with returns R = (R 1,..., R n ) in order to maximize the expected value or utility of his payoff (e.g. induced by his incentive system). The optimization is subject to a set of portfolio constraints, such as P = {x [0, 1] n x 1 n = 1}. In practice, one seldom has full information about the joint distribution of asset returns, which are typically assumed multi-variate normal and interpolated from very few data points (hence are not robust). Very often the only reliable data consists of first and second order moments. For example, RiskMetrics (www.riskmetrics.com) provides very large data sets of asset covariances, but no additional distributional information. In this case, if the manager has access only to the mean and covariance matrix of the asset returns, and not to the entire distribution, he may be interested in the best and worst case return of a given portfolio, and in the portfolios that would maximize such a worst case return. For instance, a risk neutral manager incentivised by a bonus for performance above a given target return will maximize the worst case probability of reaching that target level. There is a considerable amount of literature on robust portfolio optimization; see for example Goldfarb and Iyengar (2003), Rustem et al. (2000), El Ghaoui et al. (2003). We contribute to this stream by studying the portfolio example from the manager s perspective in Section 5. One can think about Problem (G) as a strategic game, where the first player chooses a decision x while nature picks the worst possible distribution R (see e.g. Dupačová 1966). The robust approach adopted in this paper is very much in the spirit of the max-min expected utility framework axiomatized by Gilboa and Schmeidler (1989). The concept of robustness, or incomplete knowledge about distributions, is captured in various ways in the stochastic optimization literature, the most common ones being uncertainty sets (Ben-Tal and Nemirovski 2002) and moments of the underlying distribution. Our model falls in the second category, and is related in that sense to a large literature including Scarf (1958), Birge and Wets (1987), Birge and Dulá (1991), Kall (1988), Dupačová (2001), Shapiro and Kleywegt (2002). For a given vector x, we refer to the problem (I) U(x) = min E[u(x R)] (2) R (µ,σ) as the inner optimization problem and to the function U(x) as the robust objective (or criterion). There are three main steps in our solution to Problem (G).
4 4 Operations Research 00(0), pp , c 0000 INFORMS 1. First, we transform the inner optimization problem into an equivalent problem over univariate distributions with a given mean and variance. This reduces the original problem to a bi-criteria optimization problem. 2. Second, we show that for some general classes of objective functions, this bi-criteria objective reduces to a deterministic optimization problem in one variable. 3. Finally, based on these results, we provide conditions to reduce the original problem to solving a parametric quadratic problem. The first key result of our derivations is a new projection property for classes of multi-variate distributions R with given mean vector µ and covariance matrix Σ. Based on this result, we show in Proposition 1 that for any vector x, Problem (I) reduces to an optimization problem over the class of projected univariate distributions r x with given mean µ x = x µ and variance σ 2 x = x Σx, that is: U(x) = min R (µ,σ) E[u(x R)] = min E[u(r x )]. r x (µ x,σx 2) The above problem is further reduced (Proposition 2) to optimizing over a restricted class of distributions with at most three support points. The robust objective U(x) can be cast as a deterministic optimization problem in at most three variables. This procedure reduces the robust stochastic optimization problem (G) to a deterministic bi-criteria problem that only depends on the mean and variance of the random variable x R. The results of Section 2 rely on the classical theory of moments (see e.g. Karlin and Studden 1966, Rogosinsky 1958) and require no assumptions on the objective function u. Section 3 provides conditions on the objective function u to express the robust objective U as an optimization problem in one variable. This holds for so-called one or two-point support functions u (see Definitions 1 and 3). Such functions are supported from below by a quadratic and intersect it at exactly one or two points. Special cases include concave utilities with convex, concave or concave-convex derivatives, in particular corresponding to risk averse and prudent/imprudent preferences, as well as some target-based piecewise polynomials. The resulting second order moment bounds are refinements of the Jensen and Edmundson-Madansky inequalities (see Dulá and Murthy 1992, Dokov and Morton 2002) and extend results of Krein and Nudelman (1977) and Birge and Dulá (1991) for distributions with unbounded support. These results are also useful for sensitivity analysis in many decision sciences applications (see e.g. Smith 1995), in particular in stochastic programming when the underlying distribution is actually known, but difficult to work with (Birge and Wets 1987).
5 Operations Research 00(0), pp , c 0000 INFORMS 5 In Section 4, we use results from Geoffrion (1967a) to show (Proposition 9) that for some general classes of objective functions, the robust Problem (G) over a convex polyhedral set P is equivalent to solving the following parametric quadratic program (PQP): (P λ ) max x P λ x µ (1 λ) x Σx. (3) In particular, this is true for increasing and concave utilities u that satisfy the one or two-point support property characterized in Section 3, which includes utilities with convex or concave-convex derivative. Section 5 illustrates our results with an application in the context of portfolio management incentives. We provide optimal robust portfolios corresponding to various (non-continuous, nondifferentiable) reward systems for portfolio managers, involving bonuses, commissions and combinations thereof, as well as non-expectation criteria such as value at risk. Our results show that the resulting robust portfolios are mean-variance efficient and solvable as PQPs. We obtain new results, as well as new proofs and insights for known technical results. In particular, we find that under bonus systems, a manager who assumes joint normal return distributions chooses the same portfolio as the most pessimistic manager, with the same given mean-covariance information. In Section 6 we propose two extensions of our results motivated by an application in multiproduct pricing. First, we show that our general model and solution naturally extend to the case when the distribution R (its mean and covariance) is dependent on the decision variable x. Second, we attempt to extend our framework to the case of non-negative random returns. While we prove that bi-criteria solutions are generally not optimal, we provide a method to compute (weak) bicriteria bounds on the corresponding robust problems, and show how they can be solved using parametric quadratic programming. Our conclusions are summarized in the last section. 2. The Robust Objective In this section we focus on Problem (I): for a non-zero vector x, we consider the problem U(x) = min E[u(x R)]. We show that this problem can be formulated as a bi-criterion objective, which R (µ,σ) can be cast as a deterministic optimization problem in (at most) three variables. The results of this section hold without any assumptions on the objective u A General Projection Property A lower bound on the robust objective U(x) can be obtained by optimizing over univariate distributions with mean µ x and variance σ 2 x: U(x) = min R (µ,σ) E[u(x R)] min r (µ x,σ 2 x ) E[u(r)]. (4)
6 6 Operations Research 00(0), pp , c 0000 INFORMS This is because for any feasible random vector R (µ, Σ) on the left hand side, the corresponding projected random variable r x = x R has mean x µ = µ x and variance x Σx = σx, 2 hence r x is feasible on the right hand side. Our first key result is that, irrespective of the objective function u, relation (4) above holds with equality. Proposition 1. For any objective function u and any given real vector x, we have that: min R (µ,σ) E[u(x R)] = min E[u(r)]. (5) r (µ x,σx 2) The central idea behind Proposition 1 is that for any non-zero vector x, the optimization over random vectors R (µ, Σ) is equivalent to solving the corresponding projected problem over the class of univariate distributions with mean µ x and variance σx. 2 We provide a constructive proof for the following stronger result: for any non-zero vector x and any distribution r (µ x, σx), 2 there exists a multivariate distribution R (µ, Σ) such that r = x R. We remind the notation M n for (µ,σ) the set of probability measures on R n with mean µ and covariance Σ (with the default superscript n = 1). Proposition 1 is a direct corollary of the following stronger result, proved in the Appendix: Theorem 1 (General Projection Property). For any given n-vector µ, (n n)-matrix Σ 0 and any non-zero n-vector x, the x-projection mapping r = x R from n-variate distributions R (µ, Σ) to univariate distributions r (µ x, σx) 2 is onto, meaning that every distribution r in M (µx,σx 2) can be obtained from some distribution R in M n by r = (µ,σ) x R Equivalent Deterministic Formulation Based on Proposition 1, the general robust stochastic optimization problem (G) amounts to maximizing U(x) = min E[u(r)], over the domain x P. This is a special case of a univariate moment r (µ x,σx 2) problem, where only the first and second order moments (mean and variance) of the distribution are known. In this section we show that such problems can be reduced to a deterministic optimization problems in three variables. A classic result in the theory of moments, apparently due to Rogosinsky (1958, Theorem 1), is that moment problems with n moment constraints can be solved by optimizing only over discrete distributions with at most n + 1-support points (see Smith 1995, for a more accessible presentation). In our case, this result implies that the robust objective reduces to optimizing over (µ x, σx)- 2 distributions with at most three support points. We obtain the following deterministic formulation in three variables:
7 Operations Research 00(0), pp , c 0000 INFORMS 7 Proposition 2. For any objective u and x 0, the robust objective U can be calculated as: where U(x) = min p a u(a) + p b u(b) + p c u(c) s.t. a µ x σ2 x b µ x + σ2 x c µ x µ x a c, (6) p a = { σ 2 x +(µ x b)(µ x c) (a b)(a c), if a < b 1 σ2 x +(µx a)2, if a = b, p c = (c a) 2 { σ 2 x +(µ x a)(µ x b), if b < c (c a)(c b) 1 σ2 x +(µx c)2, if b = c, p b = 1 p a p c. (c a) 2 Proof: It remains to show that if a distribution with at most three support points satisfies the (µ x, σx)-moment 2 constraints, then it is of the above form. Note that two-point distributions with mean µ x and variance σx 2 1 p assign mass p to the point µ x + σ p x and mass 1 p to the point µ x p σ 1 p x, with p (0, 1). When a = b or b = c above, then we obtain precisely two point distributions of this form. Suppose now that a < b < c. The expressions for the probabilities are equivalent to the meanvariance conditions. In order for these to be non-negative, it is necessary and sufficient that a µ x σ2 x c µ x b µ x + σ2 x c. µ x a In particular, optimizing only over feasible two point distributions in problem (6), we obtain the following upper bound on U(x): U(x) min p u(µ x + p (0,1) 1 p 3. Simplified Formulation of The Robust Objective p p σ x) + (1 p) u(µ x 1 p σ x). (7) This section investigates the case when the above upper bound is sharp. We provide conditions on the objective u for the optimum in Problem (I) to be achieved exactly or asymptotically by two point distributions. This holds for a very general class of functions u (so-called one or two-point support functions), including all monotone convex functions and all concave functions with convex, concave or concave-convex derivative, as well as some important piecewise linear classes. These results are important because they allow to reduce the robust objective to an optimization problem in one variable. Moreover, in certain cases, closed form formulations of the robust objective can be obtained based on these results, as illustrated in Section 5. Finally, in the next section, we show that optimizing Problem (G) is relatively easy for one and two point support objectives. The key element of this section is the family Q = {(A, B, C) q(y) = Ay 2 + By + C u(y) y R} corresponding to coefficients of quadratic functions dominated by u. Clearly, E[u(r)] E[q(r)] = A(µ 2 x + σ 2 x) + Bµ x + C for any random variable r (µ x, σ 2 x), implying:
8 8 Operations Research 00(0), pp , c 0000 INFORMS Proposition 3. We have U(x) = min E[u(r)] r (µ x,σx 2) max (A,B,C) Q A(µ 2 x + σ 2 x) + Bµ x + C. Strong duality for moment problems (see e.g. Karlin and Studden 1966) insures that under certain regularity conditions (in our case amounting to σ x > 0) equality holds in Proposition 3. We mention this result for its general interest, however it will not be necessary for our work Two Point Support Objectives Consider (A 0, B 0, C 0 ) Q so that q 0 (y) = A 0 y 2 + B 0 y + C 0 intersects u at two points a < b. Suppose that there exists a distribution r (µ x, σ 2 x) with support {a, b}. It follows that E[u(r)] = E[q 0 (r)] = A 0 (µ 2 x + σ 2 x) + B 0 µ x + C 0. From Proposition 3, this must equal U(x). So in this case U(x) can be obtained by minimizing only over feasible two point distributions, i.e. equality holds in (7). We formalize this result as follows: Definition 1. A function u satisfies the two-point support property with respect to (µ, σ 2 ) if there exists a quadratic function that supports u from below and intersects it at two points a, b such that a feasible (µ, σ 2 ) distribution with support {a, b} exists. A function u satisfies the two-point support property if it satisfies it with respect to any (µ, σ 2 ). Proposition 4. If u has two point support, then the robust objective equals: 1 p p U(x) = min U(p, x), where U(p, x) = p u(µ x + p (0,1) p σ x) + (1 p)u(µ x 1 p σ x). (8) A similar result for a special piecewise linear objective and non-negative distributions has been derived by Scarf (1958), in the context of an inventory problem. Birge and Dulá (1991) use two point support functions to obtain the above result over a bounded domain. Note that a function can have two-point support on an interval, but not on the whole real line, e.g. if a support point lies on the boundary. This can be seen from the following example: Example 1. Let u(y) = 1 (µ kσ,µ+kσ) (y), where 1 S denotes the indicator function of a set S. We have that min E[u(r)] = r (µ,σ 2 ) min r (µ,σ 2 ) P ( r µ < kσ) = max(0, 1 1 k 2 ). This is the two-sided Chebyshev inequality, achieved by the three point distribution with mass 1, 1 1, respectively 1, at the points µ kσ, µ, respectively µ + kσ. Note that minimizing 2k 2 k 2 2k 2 ( over two point distributions yields a strictly larger bound, namely max 1 k, 2 1+k 2 1+k 2 ). In light of Proposition 4, u cannot have two point support. On the other hand, the same objective function, defined say on [µ kσ, ) has two point support for k 1.
9 Operations Research 00(0), pp , c 0000 INFORMS 9 Denote the two support points in relation (8) by a = µ x p σ 1 p 1 p x and b = µ x + σ p x (so a < b unless x = 0). For differentiable functions u we obtain: U p (p, x) = u(b) u(a) (b (b) + u (a) a)u. (9) 2 The first order condition for U(p, x), x 0, can be stated as: u(b) u(a) b a = u (a) + u (b). (10) 2 The optimum in (8) is either obtained at a point where the above equality holds, or as p 0, or p 1. In the first case, the two point support is precisely {a, b}. In the latter cases it degenerates into the one-point support case discussed in the next section. The remainder of the section provides conditions for a function to have two-point support. The next result is a technical restatement of Definition 1: Lemma 1. The function u has two point support if and only if for any µ and any σ > 0 there exist a < b and q a, q b such that the following conditions hold: (a) (b µ)(µ a) = σ 2 ; (b) u(b) u(a) b a = q a + q b ; 2 (c) q(y) = Ay 2 + By + C u(y), y R, where A= q b q a, B= bqa aq b, C = bu(a) au(b) 2(b a) b a b a ab qa q b 2(b a). Proof: From (b) and (c), it follows that q(a) = u(a) and q(b) = u(b) and q supports u. Condition (a) states that the intersection points a, b are the support of a (µ, σ 2 )-distribution. Hence the conditions of Definition 1 are met. Conversely, if u has two point support, then the conditions of the lemma are met for q a = q (a) and q b = q (b), where a, b and the quadratic q come from Definition 1. By Proposition 3, the above result shows that: Remark 1. If u satisfies the conditions of Lemma 1, then an optimal distribution for U(x), if it exists, has support {a, b} as defined by the conditions of the Lemma with respect to (µ x, σ x ). If u is differentiable at a, b, the construction in Lemma 1 implies q a = q (a) = u (a), q b = q (b) = u (b), i.e. q intersects u tangentially at a, b. In this case, condition (b) in Lemma 1 becomes precisely Eq. (10). This condition says that the average value of the derivative u (if it exists) on [a, b] equals the average of the values of u at the end points a, b. Geometrically, the area under the graph of u between a and b equals the area of the trapezoid with corners a, u (a), u (b), b. The existence of two such points a, b is insured for example if u is non-increasing and changes convexity at least once (between a and b). Moreover, we show that conditions (a) and (c) are satisfied if u is concave until some point and then convex, and has finite limits at ±.
10 10 Operations Research 00(0), pp , c 0000 INFORMS Definition 2. A function f defined on R is convex-concave if there exists x 0 R such that f is convex on (, x 0 ) and concave on (x 0, ). A function f defined on R is concave-convex if f is convex-concave. A function f is S-shaped if it is increasing and convex-concave. A function f is inverse S-shaped if f is S-shaped. The following corollary of Lemma 1 is proved in the Appendix. An analogue result has been obtained by Birge and Dulá (1991, Theorem 5.1) for functions with bounded support. Proposition 5. If u is differentiable such that u is inverse S-shaped with finite limits at ±, then u satisfies the two-point support property. Furthermore, if u has two point support, then by Definition 1, u(x) + q(x) also has two point support for any quadratic q. Examples of functions satisfying these conditions include combinations of exponentials and logarithms (e.g. log-logistic u(x) = C + log 1/(1 + e ax ) used in statistics, classification), and hyperbolic functions (e.g. the catenary u(x) = C b cosh(ax) used in physics, engineering), where in all cases the constant C can be replaced by a concave quadratic. The conditions of Proposition 5 are not necessary, however, for a function to have two-point support. In general, two-point support functions need not be differentiable, not even continuous; such examples are investigated in Section 5. Furthermore, even for differentiable functions neither concavity, nor the concave-convex derivative condition are necessary for two-point support, as illustrated in the following counterexample. Example 2. The function u(y) = y ln(1 + y 2 ) is increasing, but not concave (a plot of its derivative is provided in Figure 1). However, u has the two-point support property. Note that u (y) + u ( y) = 2 for all real y. Hence conditions (a) and (b) of Lemma 1 hold for b = a = µ2 + σ 2 and q a = u (a), q b = u (b). By construction, the quadratic q defined in part (c) of the Lemma intersects u tangentially at a, b. The convexity properties of u imply that q supports u. From Proposition 4, for a given vector x, we have that U(x) = min U(p, x), where p (0,1) 1 p U(p, x) = µ x p ln(1 + (µ x + p σ x) 2 ) (1 p) ln(1 + (µ x p 1 p σ x) 2 ). Remark that lim U(p, x) = lim U(p, x) = u(µ x ) = µ x ln(1 + µ 2 p 0 p 1 x), and U(p, x) is decreasing in p around p = 0 and increasing around p = 1, so the minimum is achieved somewhere in-between. Suppose, for example that for a certain x = x 0 we have µ x = 0 and σ x = 1. Then U(p, x 0 ) = p ln p + (1 p) ln(1 p), which is minimized for p = 1/2 and U(x 0 ) = ln 2.
11 Operations Research 00(0), pp , c 0000 INFORMS 11 Figure 1 Plots of the functions u(y) = y ln(1 + y 2 ) and u (y) = 1 2y 1+y u 2 u One Point Support Objectives The previous section provided a simplified expression for the robust objective U(x) when u has twopoint support. In particular this is true when u is concave with concave-convex derivative. We now provide other conditions that allow a simplified formulation for U when u has one point support. Relevant special cases include u concave with a convex or concave derivative, and u monotonic and convex. The one-point support property is a limiting case of two-point support, where the expected value of the supporting quadratic is approached by a sequence of feasible two-point distributions, with one point approaching the mean and the other arbitrarily large (or small). This is formally defined as follows: Definition 3. A function u has the one-point support property with respect to (µ, σ 2 ) if there exists a quadratic q that supports u and intersects it at µ, and either lim E[u(r p )] = E[q(r p )] p 0 or lim E[u(r p )] = E[q(r p )], where r p is a (µ, σ 2 1 p ) random variable with support {µ + σ, µ p 1 p p σ}. A function u has the one-point support property if it satisfies it with respect to any 1 p (µ, σ 2 ). The following result follows trivially from Proposition 3. Proposition 6. If u satisfies the one point support property then U(x) = min U(p, x), specifically p (0,1) either U(x) = lim U(p, x) or U(x) = lim U(p, x). p 0 p 1 In the following we identify some general relevant classes of one-point support functions. Proposition 5 shows that concave functions with concave-convex derivative have two-point support. In the extreme case when the derivative is either convex or concave (x 0 = ± in Definition 2), i.e.
12 12 Operations Research 00(0), pp , c 0000 INFORMS the risk averse agent is prudent or imprudent, we obtain one-point support functions. The proof of the next result is provided in the Appendix. Proposition 7. If u is concave and twice differentiable with a monotonic second derivative then u has one-point support. Specifically, (a) if u is convex, then U(x) = u(µ x ) + lim y u (y) σ2 x 2 (b) if u is concave, then U(x) = u(µ x ) + lim y u (y) σ2 x 2 for all x; for all x. Example 3. Consider a portfolio manager facing an exponential utility u(x) = 1 e ax, with a > 0. Since lim y u (y) =, Proposition 7(a) shows that the robust solution should focus on maximizing the expected portfolio return µ x while setting σ x = 0, i.e. finding a perfect hedge. In particular, if Σ is non-singular, the optimal robust solution is not to invest at all, as for any investment strategy there exists a corresponding distribution that can make the expected utility of the return arbitrarily small. (These are two point distributions, with an arbitrarily large loss probability 1 p 1). While this is an extreme scenario, one should expect the robust approach to yield strategies that are by definition conservative. We now consider the case when u is monotonic and convex, so we need to optimize over concave quadratics that are tangent to u at exactly one point. The next result is proved in the Appendix. Proposition 8. If u is monotone convex, then it has one point support and U(x) = u(µ x ). More generally, the same result holds if u is convex with either u ( ) = 0 or u ( ) = 0. The class of one-point support functions includes utility functions most commonly used for mathematical modeling: those whose (first three) derivatives alternate in sign, i.e. those consistent with (up to third order) stochastic dominance (see Brockett and Golden 1987). Examples include the standard exponentials, logarithms, power functions and hyperbolic functions (see Pratt 1964), sumex and linear plus exponential (Bell s one-switch functions, see Bell 1988), or more generally, any mixtures of exponentials. In summary, one or two-point support functions are fairly general, including (but not limited to) convex monotone functions and concave functions with at most one third derivative sign change. Some additional relevant non-smooth classes include piecewise linear functions, discussed in the context of a financial application in Section Bi-criteria Optimization via PQP The previous sections focused on reducing the inner problem from an optimization problem over multivariate distributions to a deterministic optimization problem in one or three variables. The remaining step in the solution to Problem (G) is to optimize this robust objective.
13 Operations Research 00(0), pp , c 0000 INFORMS 13 This section exploits the fact that the robust objective depends on the decision variable x only through two criteria: the return s mean µ x and standard deviation σ x, U(x) V (µ x, σ x ). This allows a simple and efficient parametric quadratic programming (PQP) formulation and solution to Problem (G), based on Geoffrion (1967a). For a fairly general class of functions u, we show that optimizing the corresponding robust objective is equivalent to a mean-variance optimization problem, for a certain trade-off coefficient, i.e. the resulting optimal solution of Problem (G) is mean-variance efficient. Based on Propositions 4 and 6, we argue that these results hold in particular for increasing and concave utilities with one or two-point support, as well as other non-concave objectives, as illustrated in Section 5. Recall that a real function f defined on a convex set S is said to be quasi-concave if and only if for any λ [0, 1] and x, y S we have f(λx + (1 λ)y) min(f(x), f(y)). Equivalently, f is quasi-concave whenever for any l, the upper level sets L l = {x S f(x) l} are convex. The following result due to Geoffrion (1967a) (see also Geoffrion 1967b) provides general sufficient conditions for solving bi-criteria problems efficiently as parametric quadratic programs. Theorem 2 (Geoffrion 1967a). Consider the program max v(p 1(f 1 (x)), p 2 (f 2 (x))), (11) x X where X R n is a convex set. Suppose that f 1, f 2, p 1 (f 1 ), p 2 (f 2 ) are concave on X, p 1 and p 2 are increasing on the image of X under f 1, f 2 respectively, and v is non-decreasing in each coordinate and quasi-concave on the convex hull of the image of X under (p 1 (f 1 ), p 2 (f 2 )), and all functions are continuous. Further assume that the optimal solution x (λ) of the parametric program max x X λf 1(x)+ (1 λ)f 2 (x) is continuous for λ [0, 1]. Then the function w(λ) = v(p 1 (f 1 (x (λ))), p 2 (f 2 (x (λ)))) is continuous and unimodal on [0, 1]. Furthermore, if λ is its maximum, then x (λ ) is optimal for (11). Proposition 1 reduced Problem (G) to solving a bi-criteria (mean-variance) optimization problem. Based on Theorem 2, the following result shows that under fairly unrestrictive conditions, Problem (G) can be optimized by solving a parametric quadratic program. Proposition 9. Suppose that the robust objective function V (µ x, σ x ) = min E[u(r)] is continuous, non-decreasing in µ x, non-increasing in σ x and quasi-concave. Then Problem (G) over r (µ x,σx 2) a convex polyhedral set P is equivalent to solving the following parametric quadratic program: (P λ ) max x P λ x µ (1 λ) x Σx. (12)
14 14 Operations Research 00(0), pp , c 0000 INFORMS Specifically, if x (λ) denotes the optimal solution of Problem (P λ ), then the function w(λ) = V (µ x (λ), σ x (λ)) is continuous and unimodal in λ [0, 1]. Moreover, if λ x (λ ) is optimal for Problem (G). is its maximum, then Proof: Let f 1 (x) = µ x and f 2 (x) = x Σx, p 1 (f 1 ) = f 1 and p 2 (f 2 ) = ( f 2 ) 1 2, these are all concave, and the latter two are non-decreasing. In the notation of Theorem 2, we obtain v(µ x, σ x ) = v(p 1 (f 1 (x)), p 2 (f 2 (x))) = min E[u(r)] = V (µ x, σ x ). The assumptions on V insure that v is nondecreasing and quasi-concave. Furthermore, the convex polyhedral structure of the feasible set r (µ x,σx 2) P insures that there exists a continuous optimal solution x (λ) (see e.g. Wolfe 1959, Geoffrion 1967a). The result follows from Theorem 2. It is reasonable to assume that the robust utility U(x) = V (µ x, σ x ) is non-decreasing in the mean µ x and non-increasing in the standard deviation σ x of the underlying return. This basically says that the decision maker prefers distributions with higher return and lower risk. Quasi-concavity can be interpreted as a diminishing marginal rate of substitution between mean and standard deviation, a property shared by most utility functions. When the actual utility function u is increasing and concave (as is typically assumed in decision analysis), and has one or two-point support, then these conditions are automatically satisfied. We obtain the following result: Proposition 10. For increasing concave utilities with one or two point support, Problem (G) over a convex polyhedral set P is equivalent to solving the parametric quadratic program (P λ ) for a suitable value λ [0, 1]. Proof: From Propositions 6 respectively 4, we have that V (µ x, σ x ) = min V (p, µ x, σ x ), where p (0,1) 1 p p V (p, µ x, σ x ) = p u(µ x + p σ x) + (1 p) u(µ x 1 p σ x). Monotonicity of u implies that V (p, µ x, σ x ) is non-decreasing in µ x. Concavity of u implies monotonicity of V (p, µ x, σ x ) in σ x, and concavity in (µ x, σ x ). Since monotonicity and concavity are preserved through minimization, the same results hold for V (µ x, σ x ). The final result follows from Proposition 9. In particular, the above results hold for increasing concave utilities with convex or concaveconvex derivative. For more general utilities, one can attempt to check monotonicity of V in µ x and σ x as well as quasi-concavity using the results of Propositions 2, 4 or 6, depending of the support properties of the function u. In particular, for one or two-point support functions, monotonicity in µ x can be ensured by assuming that u is non-decreasing. Some examples are provided in the next section.
15 Operations Research 00(0), pp , c 0000 INFORMS 15 Remark that if u is concave and has one or two point support, Propositions 4 and 6 imply that U(x) is concave in x. So alternatively, traditional computational methods for convex programming may also be suitable to solving Problem (G) without exploiting its bi-criteria structure. Several algorithms are available for solving parametric quadratic programs (Wolfe 1959, Geoffrion 1967a,b, Best 1996). They all yield an optimal solution x (λ) that is continuous in λ. Geoffrion (1967a) solves (P λ ) with λ = 1 and decreases λ until the unimodal function w (λ) = v(µ x (λ), σ x (λ)) achieves its maximum on [0, 1]. In our applications, the intermediate solutions are also of interest, as they yield a portion of the mean-variance trade-off curve. More recent algorithms for PQP are due to Best (1996) and Arseneau and Best (1999). For a general treatment of parametric quadratic programs see Bank et al. (1982). Sensitivity analysis for quadratic programs in the mean-variance framework of Markowitz (1959) is discussed in Dupačová et al. (2002, Ch. II.7). Example 4. In Problem (G), suppose that the constraint set is given by P = {x x i = 1, x 0}, and let u(y) = ln( 1 1+e y ). Then u is increasing and concave and its derivative u (y) = e y 1+e y is inverse S-shaped (see Figure 2). Therefore V (µ x, σ x ) = min p ln(1 + e x+q (µ p (0,1) 1 p p σ x) ) (1 p) ln(1 + e (µx p 1 p σx) ) is increasing in µ x, decreasing in σ x and concave in (µ x, σ x ), i.e. satisfies the conditions of Proposition 9. To solve Problem (G), we first compute x (λ), the optimal solution of: (P λ ) max λ x µ (1 λ) x Σx s.t. xi = 1 x 0, and then maximize over λ (0, 1) the function w(λ) = V (µ x (λ), σ x (λ)), which is unimodal. This can be seen from Figure 3, where we display w(λ) for the following three data sets: µ 1 = (1 3), Σ 1 = ( ) , µ = ( ) , Σ 2 = , respectively µ 3 = µ 2, Σ 3 = Σ 2. The resulting optimal values w(λ ) are the desired optimal values of (G), in our case , and , achieved at λ =.60,.72, and respectively 1. The corresponding optimal solutions of Problem (G) are x (λ ) = ( ), ( ) and ( ). All programs were run in MATLAB 5.3 under Windows XP. 5. Application: Portfolio Management Incentives In this section we present an applications of our results in the context of financial portfolio management incentives. This application prompts us to analyse a set of special non-continuous/nondifferentiable two-point support objectives, induced by bonus and commission-based incentives, as
16 16 Operations Research 00(0), pp , c 0000 INFORMS Figure 2 Plots of the functions u(y) = ln( ) and u (y) = e y 1+e y 1 1+e y. u u Figure 3 Plots of the unimodal function w(λ) w 1 (λ) 0.4 w 2 (λ) 0.3 w 3 (λ) λ λ λ well as non-expectation criteria such as value at risk. We obtain new results, as well as new proofs and insight for some technically known results. Consider the example described in the introduction: a portfolio manager is interested in a robust portfolio among a set of n assets with random returns R 1,..., R n. Suppose that the manager does not know the distribution of the random returns, but estimates them to have means µ and covariance structure Σ. The manager is interested in a robust portfolio that maximizes the worst case expected payoff u(r), where r is the portfolio return achieved at the end of the evaluation period. This problem can be formulated as follows: max x P min R (µ,σ) E[u(x R)], where the underlying feasible set P is convex polyhedral, and may have different forms, according to the type of portfolio problem of interest. For example, P = {x R n A x b} or in particular P = {x [0, 1] n x 1 n = 1}. The objective function u can either represent the agent s utility function for a reward proportional to the portfolio return, or the valuation induced by a more complex
17 Operations Research 00(0), pp , c 0000 INFORMS 17 compensation mechanism. For example, if a (risk neutral) agent is rewarded a one-shot bonus for outperforming a certain index s, then u is precisely the cdf of s. According to Proposition 10, if u is increasing and concave (i.e. the agent is risk averse) with one or two-point support, this problem amounts to solving the PQP (12) over the portfolio choice set P for a suitable value λ. In particular, the resulting optimal portfolio is mean-variance efficient (a portfolio is mean-variance efficient if it has the highest expected return for a given variance, or, equivalently, if it has smallest variance for a given expected return (see e.g. Markowitz 1959). The robust approach associates with each utility function u a worst case risk factor κ = (1 λ )/λ, obtainable by the methods of Theorem 2 and Proposition 9, such that solving the robust optimization problem is equivalent to solving a deterministic problem with quadratic utility u κ (x) = µ x κ x Σx. This is a deterministic mean-variance portfolio problem that can be solved by standard methods. Consider for comparison purposes the alternative approach, based on the assumption that the underlying distribution of returns is multivariate normal. In this case the problem can be formulated as follows: (N) max E[u(r x)] = max x P x P 1 u(y) e (y µ x) 2 x Σx dy. 2π For increasing and concave utilities u, it is well known that solving the above problem yields a mean-variance efficient portfolio (see e.g. Kira and Ziemba 1977). Moreover, for exponential utilities, the solution can be easily computed. For more general utilities, however, Problem (N) involves numerical integration. In the following we investigate Problem (G) for some relevant non-differentiable criteria that arise from typical incentive schemes for portfolio managers. The resulting objectives do not satisfy the conditions of Proposition 10, so stronger results are needed. We first argue that the objective has two point support, then calculate the robust objective U(x) = V (µ x, σ x ) in closed form. Finally, we show that V satisfies the conditions of Proposition 9, hence the optimal robust portfolio is mean-variance efficient and the problem can be solved as a PQP. Bonus and Commission Incentives. Portfolio managers, and generally sales managers, are typically incentivised by a fixed salary plus a bonus β, contingent on the realization of a certain preset performance target, denoted α. It is often argued that fixed target bonus systems are unappealing because managers have no direct incentive to over-perform the target. As a remedy, managers are offered an additional commission rate γ for over-achieving the target. Such a reward system, mixing a bonus β and commission rate γ for output above the target α, can be modeled
18 18 Operations Research 00(0), pp , c 0000 INFORMS as u(y) = γ(y α) + + β1 (y>α). If the distribution of returns is normal, then the problem amounts to optimizing (numerically) the objective: γσ x Ψ( µ x α ) + βφ( µ x α ), where Ψ(a) = E[(Z + a) + ], Φ(a) = P (Z a), Z N(0, 1). (13) σ x σ x Given only means and covariances of the asset returns, the manager maximizes over x P the corresponding robust criterion: U + α (x) = min R (µ,σ) γe[(x R α) + ] + βp (x R > α) = min γe[(r α) + ] + βp (r > α), (14) r (µ x,σx 2) where the last equality follows by Proposition 1. The following result is proved in the Appendix. Proposition 11. The optimal robust portfolio for a manager incentivized by a bonus β plus a commission rate γ for performance above a target α maximizes over x P the robust objective: (µ U + x α) 2 + α (x) = γ(µ x α) + + β σx 2 + (µ x α). 2 Moreover, the optimal portfolio is mean-variance efficient and can be solved as a PQP. In particular, with a commission-only reward (β = 0), the manager only cares about expected return. If the manager is only rewarded a bonus (γ = 0), the objective is to maximize the (worst case) probability of achieving the given target α. Proposition 11 reduces the optimal robust portfolio to maximizing (µ x α) +. This follows also from Proposition 1 and Chebyshev s inequality. For σ x comparison, from (13), the corresponding problem with normal returns is equivalent to optimizing the portfolio s Sharpe ratio µ x α σ x (see e.g. Charnes and Cooper 1963). The interesting analogy between the robust and normal criteria can be interpreted as follows: Consider two portfolio managers who have access to the same set of assets with known means µ and covariance Σ and are incentivised by the same achievable target α (i.e. there exists x P such that µ x α). The first manager believes that stocks follow a multivariate normal distribution with these parameters, whereas the other manager ignores the distribution and adopts a worst case approach. This result says that the two managers, despite their different beliefs, will choose the same portfolio. Penalty and Value at Risk (VaR) Incentives. One problem with the above incentives is that managers who are not on the winning track (i.e. have low chances of making the bonus) may undertake extremely risky strategies, with potentially disastruous consequences for the firm. As a remedy, down-side incentive schemes are designed to minimize risk exposure; we consider two
19 Operations Research 00(0), pp , c 0000 INFORMS 19 examples, based on penalties and value at risk. With both up and down-side incentives in place, successful managers will attempt to maximize gains, while under-performers minimize losses. Ross (2004) suggests an incentive system whereby managers are penalized for performances below a benchmark level α; this is modeled by an option-type payoff u(y) = (α y) +, with the unit penalty normalized wlog to 1. The corresponding robust portfolio minimizes over x P the worst case penalty Uα = max R (µ,σ) E[(α x R) + ]. Although this is a min-max formulation, a simple change of sign transformation can convert it to max-min, and our standard results apply. The next result is proved in the Appendix. A direct proof of the univariate bound is due to Jagannathan (1977), who also provides solutions for non-negative and symmetric distributions. Proposition 12. The optimal robust portfolio for a manager incentivized by a linear penalty for performance below a target α, minimizes over x P the robust criterion: U α = 1 2 [(α µ x) + σx 2 + (α µ x ) 2 ]. Moreover, the optimal portfolio is mean-variance efficient and can be solved as a PQP. Another relevant measure of downside risk, which financial firms, as well as portfolio and insurance managers are often required to quote and optimize is value at risk (see e.g. Linsmeier and Pearson 1996). The value at risk of a return x (or loss x) at a confidence level α (typically α = 0.95 or α = 0.99) is defined as the following fractile criterion: V ar(α; x) = min{ξ P ( x < ξ) α} = max{z P (x > z) α}. The definition with strict inequality is needed for our results. However, weak inequalities can replace strict ones throughout if return distributions are known to be continuous. For normal returns, Charnes and Cooper (1963) prove that V ar N (α) = Φ 1 (α)σ x µ x. In practice, return distributions are typically not normal, and exact computation of VaR may amount to difficult numeric integration. Robust (or worst case) value at risk is defined as the largest VaR attainable, given only the mean and covariance of the returns. The robust portfolio with VaR criterion solves: V ar(α) = max x P {z min P (x R > z) α} R (µ,σ) = max {z (µ x z) 2 α, z µ x P (µ x z) 2 + σx 2 x }. The second formulation follows by Proposition 11 for γ = 0; the optimum is achieved when equality holds in the first constraint. Solving for z, we obtain the following result; a proof based on the multivariate Chebyshev inequality (Karlin and Studden 1966) is due to El Ghaoui et al. (2003). (15)
20 20 Operations Research 00(0), pp , c 0000 INFORMS Proposition 13. The optimal robust portfolio for a manager minimizing an α-var incentive α solves min x P 1 α σ x µ x, so it is mean-variance efficient. Notice that the VaR objective is not an expectation; this suggests possible extensions of our results beyond the main framework of this paper. 6. A Multi-Product Pricing Application and Extensions In this section we consider the optimal pricing decision of a firm (or sales manager) setting prices for a line of products so as to maximize the expected utility of profits (respectively, of his expected compensation). In this case the random returns are the demands for each product, which are non-negative and depend on the set prices. This setting indicates two possible extensions of our original problem, which we address in this section: (1) we incorporate dependence of the random vector R on the decision variable x, and (2) we analyse the case of non-negative random vectors R. We extend Proposition 9 to address the first issue, and we provide weak bounds for non-negative random variables to address the second. Consider a firm setting a vector of prices x for a set of n products with random demands R(x) = (R 1 (x),..., R n (x)) in order to maximize the expected utility of profits, given by u(r), where r is the total profit. Suppose that the demands for the n products are correlated due to substitution and/or complementarity effects, and depend on the prices of all products. Demand is estimated to have mean µ(x) and covariance structure Σ(x). The variable cost per product is given by the vector c. The problem of the firm looking for a robust pricing scheme for the n products can be formulated as follows: ( G) max x P min E[u((x c) R(x))]. R (µ(x),σ(x)) The marketing constraints on the feasible set of prices can be assumed to be of the following form: P = {x R n + A x b, x min i x i x max i } Returns Dependent on the Decision Variable The above problem ( G) is not a special case of the robust problem (G) studied in this paper, since the random return (in this case the demand), together with its mean and variance, actually depend on the decision variable (in this case the price x). While the results of this section are presented in the context of the specific pricing application, one can naturally extend them to the general abstract setting.
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Robert Bly will present a reading at Plymouth State University Sunday, March 7 at 3 p.m. The free program is a presentation of the Eagle Pond Authors' Series. It will held in the multipurpose room of the Hartman Union Building, on High Street.
New Hampshire poet laureate Cynthia Huntington says, "Bly's influence on American poetry in the last half of the 20th century has been incalculable. His poems and criticism remain alive and provocative over decades of examination. Through his own poems, his introduction of new poets, and his translations of South American and European writers he helped to reshape the landscape of poetry for several generations of readers and writers."
Bly was born in Minnesota in 1926. He attended Harvard University and received his M.A. from the University of Iowa in 1956. As the editor of the magazine The Sixties (begun as The Fifties), Bly introduced many unknown European and South American poets to an American audience. He has also edited numerous collections and nonfiction books.
Bly's honors include Guggenheim, Rockefeller, and National Endowment for the Arts fellowships.
The Eagle Pond Authors' Series is named in honor of Donald Hall, who lives at Eagle Pond Farm in Wilmot, a family homestead which has inspired much of his writing. In an interview by Francis Quinn for the April 2000 Paris Review, Bly talked about his friendship with Donald Hall and the importance of interacting with other writers. He said, "Donald Hall and I have been sending poems back and forth twice a week for 40 years. … My generation did a lot with letters. Galway Kinnell and Louis Simpson and Don and I and James Wright would often send five- and six-page typed letters commenting on and arguing with each others' poems. … The gist of it is that no one writes alone: One needs a community." Hall will introduce his friend at the reading.
Tickets for the Eagle Pond Authors' Series are free, but seating is limited. Call the Box Office at 535-ARTS for reservations. Box office hours are Monday through Friday, noon – 6 p.m. A reception and book signing follow the reading, which is presented by Plymouth State University and sponsored by the Barnes and Noble Bookstore at PSU.
The final reader in the 2003-2004 series will be Frank X. Gaspar, Sunday, April 18 at 3 p.m.

subset: commoncrawl


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Dhaka, Mar 10: The stage is ready in Bangladesh for the ICC World Twenty20 2014 tournament to be unveiled. The global ICC event will be held from March 16 to April 6 across three venues.
Schedule; Squads; Record-holder Dhoni
Here are the must-know facts about the tournament
Tournament duration - March 16 to April 6
Number of teams - 16 (increased this year from 12 to 16)
Match times - 3 PM IST (9.30 AM GMT); 7 PM IST (1.30 PM GMT); 11 AM IST (5.30 AM GMT)
Times for semi-finals and final - 6.30 PM IST (1 PM GMT)
Number of teams in qualifiers - 8
Number of teams for main draw (Super 10) - 10 (2 qualify from first round)
Tournament format - Top 8 sides from ICC World T20 2012 have got direct entry into main draw. Two will qualify from first round where six teams compete including hosts Bangladesh
Teams in Quailfying stage - 8 - Ireland, Afghanistan, Nepal, UAE, Netherlands, Hong Kong, Bangladesh, Zimbabwe
First round (qualifiers) - March 16 to 21
Qualifying groups - Group A: Bangladesh, Afghanistan, Nepal, Hong Kong. Group B: Zimbabwe, Ireland, UAE, Netherlands. Top team from each group qualifies
Qualifying matches - 12
Main draw - March 21 to April 6
Main draw pools - Group 1: Sri Lanka, South Africa, England, New Zealand, Qualifier (B1). Group 2: India, Pakistan, Australia, West Indies, Qualifier (A1)
Semi-finals: Top two from each group will make the semi-finals
Host cities - Chittagong, Dhaka, Sylhet
Stadiums - Zahur Ahmed Chowdhury Stadium (Chittagong), Sher-e-Bangla National Cricket Stadium (Mirpur, Dhaka), Sylhet International Cricket Stadium (Sylhet)
Total number of matches - 35
Number of days - 22
Semi-finals and Final - Sher-e-Bangla Stadium on April 3, 4 and 6


text:
By Alan Gomez, USA TODAY
Forty-one years after repeatedly leading a two-helicopter team into a region of Vietnam dubbed the "Valley of Death," retired Army helicopter pilot Bruce Crandall received the Medal of Honor on Monday at a White House ceremony.
Crandall, 74, might have been eligible to receive the award — the nation's highest military medal — years earlier. But he withdrew his name from consideration in the 1990s because he worried his quest for the medal might take precedence over that of his wingman and friend, Ed Freeman.
"If only one of them were to receive the Medal of Honor, he wanted it to be his wingman," President Bush said in the East Room before placing the blue-ribboned medal around Crandall's neck. "Today the story comes to its rightful conclusion: Bruce Crandall receives the honor he always deserved."
Freeman received his medal in a 2001 ceremony at the White House. Crandall, of Manchester, Wash., attended the event. Freeman, 79, couldn't attend Crandall's ceremony because his flight from his hometown in Boise was snowed in.
The two have been friends since basic training. On Nov. 14, 1965, they teamed up during a vicious battle later memorialized in a book and the Mel Gibson film We Were Soldiers.
On that day, about 450 American troops had landed in a remote area of Vietnam near the Cambodian border. They were unaware that their landing in the Ia Drang Valley was amid a major sanctuary for the North Vietnamese Army.
The Americans were surrounded by about 2,000 North Vietnamese soldiers and came under intense attack. Crandall and Freeman flew the injured out to a base and returned with troops and ammunition. They kept making trips back and forth even though they were being shot at by enemy fighters, sometimes only 30 yards away.
They flew through a cloud of smoke and a wave of bullets, Bush said. Ground commanders ordered a halt to the airlifts because of the danger, but Crandall kept going, and Freeman joined him.
"We knew that if we didn't do it our guys would go down," Crandall said in an interview last week.
The two men made 22 trips over 14 hours. Crandall switched helicopters three times after the aircraft were shot up.
The two men pulled more than 70 Americans out of the battle zone. The battalion lost 79 men, but the North Vietnamese were overcome.
"For the soldiers rescued, for the men who came home, for the children they had and the lives they made, America is in debt to Bruce Crandall," Bush said. "It's a debt our nation can never really fully repay, but today we recognize it as best as we're able."


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Space tourism [1]
NEW YORK – Future passengers aboard Virgin Galactic spaceliners can look forward to cushioned reclining seats and lots of windows during suborbital flights aboard SpaceShipTwo, a concept interior of which was unveiled by British entrepreneur Sir Richard Branson Thursday.
The air-launched SpaceShipTwo is designed to seat eight people – six passengers and two pilots – and be hauled into launch position by WhiteKnightTwo, a massive carrier craft currently under construction by Scaled Composites, Virgin Galactic president Will Whitehorn said.
For an initial ticket price of $200,000, Virgin Galactic passengers will buy a 2.5-hour flight aboard SpaceShipTwo and launch from an altitude of about 60,000 feet (18,288 meters), while buckled safely in seats that recline flat after reaching suborbital space. A flight animation depicted passengers clad in their own personal spacesuits as they reached a maximum altitude of at least 68 miles (110 kilometers).
Branson's Virgin Galactic spaceliners are slated to roll out and begin test flights by early 2008 in Mojave, California, with future operational spaceflights to be staged from New Mexico's Spaceport America beginning in 2009.
"SpaceShipTwo is obviously designed as a commercial vehicle,".
Passengers will have several minutes of weightlessness during the spaceflight, and then have about 40 seconds to return to their seats, Whitehorn said, adding that the floor of SpaceShipTwo is also designed to be used during landing of spaceflyers fail to reach their spots in time.


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Japanese Egg-Shelling Gadget For Homes
Shelling hard-boiled eggs can be quite a difficult task for some of us, and for those without nimble fingers it can be quite impossible to produce a perfect looking-one without accidently breaking the egg-white. There is a couple of ways to peel an egg manually, such as:
Rolling it on the countertop and peeling off the egg shell in a single piece;
Or the alternative method of making two holes at either end of the egg and giving it a blow to get the egg to pop out as demonstrated in the video below:
Here's one Japanese gadget spotted at Rocketnews24 that makes use of the 'blowing' method to shell a hard-boiled egg, but in a more hygienic way.
The principle is the same, which is by making two holes at both ends of the egg and then placing the egg on the stand. Next, instead of using a human mouth, something that looks like a balloon pump is placed over the egg to pop it out of its shell with air pressure.
With a good push, from the bottom out comes a shelled hard-boiled egg ready to be eaten.
Watch the Japanese egg-shelling kitchen gadget in action in the video below:
One interesting and convenient way to peel hard-boiled eggs, especially if preparing them for a large number of people at a party or gathering.


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Template:Infobox Radio station
CJRS, branded as Radio Shalom Montreal, is a 24-hour non-profit radio station based in Montreal, Canada. It is Canada's first and only trilingual Jewish radio station. It broadcasts in French, English and Hebrew. The station broadcasts at 1650 AM. Its current president is Robert Levy.
Radio Shalom is the only radio station in Canada that broadcasts mizrahi, yiddish, ladino, klezmer, hassidic and Israeli music. Radio Shalom is also, the only all Jewish radio station in North America. In accordance to Halakha (Jewish law), Jewish programming on Radio Shalom is suspended on Shabbat and on Jewish holidays. During those periods, the station brands itself simply as CJRS 1650 AM, and also as Radio CMM (Communications Michel Mathieu) and CKZW 1650, Le son gospel du Quebec. French gospel music is featured during these times.
Call sign
The station's call letters of CJRS are said to represent the station's branding of Radio Shalom. A common interpretation is that they actually stand for "Canadian Jewish Radio Station" or "Radio Shalom," and station executives have confirmed that they, in fact, may. The fact that CJRS and Radio Shalom share similar initials of RS is purely coincidental.
The CJRS call sign was previously used by Radiomutuel's Sherbrooke outlet, at 1510 on the dial. It went off the air along with a host of other AM stations in 1994.
When CJRS plays French gospel music, they identify themselves as CKZW 1650, Le son gospel du Quebec. It is unknown whether or not the CKZW call letters have been registered with the Canadian Radio-television and Telecommunications Commission.


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AURORA, Colo. – Aurora's Stanley Aviation was a leader in aerospace engineering and manufacturing for over 50 years. Now, Stanley Marketplace, a community of like-minded businesses and people, keeps innovating at the same location.
Aurora History Museum's newest exhibit, "From Manufacturing to Marketplace: Innovation at Stanley Aviation," runs March 14 through July 9 and highlights the technology and the people that made Stanley Aviation a success. The exhibit provides insight about how Stanley Aviation's history of innovation has carried on into the new Stanley Marketplace. Stanley Marketplace's redevelopment of the former aviation facility embraces the site's history in a new community-focused marketplace.
Every Sunday during the exhibit, the Aurora History Museum will feature the Robert Stanley Innovation Station, which will test museum visitors' problem-solving and innovation skills in the spirit of Stanley Aviation.
Originally founded in Buffalo, New York, in 1948 by aviation pioneer Robert Stanley, the company opened its Aurora facility at 2501 Dallas St. in 1954. Stanley Aviation designed, developed and manufactured ejection seats as well as other innovative air and aerospace technologies until it closed its Aurora facilities in 2007. Local entrepreneurs bought the property in 2014 and turned it into Stanley Marketplace, which opened its doors in late 2016.
Admission to the Aurora History Museum, 15051 E. Alameda Parkway, is always free. For more information, call 303.739.6660 or visit AuroraMuseum.org.


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This is certainly worth a look ladies. Last night in Australia there was an editorial on one of the Current Affairs shows on plus size models and the face of Australian fashion and where they anticipate fashion, models and "real women" heading this year. The story takes a little while to get going, but it does eventually punch up towards the end. The amazing City Chic was featured, but in the story it also comes to light that some other labels in OZ have just upped their size range.
Charlie Brown doesn't look to have the greatest online store - but if you are in Australia, why not head in and see what the fuss is about... apparently the store now caters to a size 16... while the amazing (and expensive) Leona Edmiston boutique now offers some designs up to a size 24. That is fabulous!
Also - keeping in Australia, there is a great article on ninemsn from last week about Aussie plus size models and their increasing demand. I love that models like Abby Valdes (pictured above) are doing so well! AND I find it heartening that fashion culture is catching up.
Take a look at the Editorial from A Current Affair - as I mention it limps at first but feels like it packs quite a punch as plus size fashion hits it's stride toward the end.
As an aside City Chic has just released a whole heap of really lovely, feminine items for the Australian Summer, like the Top "Frill Front Stripe" (RRP $49.95 AUD) - so why not get online and support fashion from down under. City Chic is my favourite Aussie plus-size store, and certainly is of an excellent quality - why not check out their sale too!
Smooch. x.


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Our rare coupon for Lip Smacker has reset! You should be able to get two more prints. Target sells these for $1.44-$1.48 each, so you can pick them up for less than a buck after coupon. Stephanie had me print this coupon, because she's thinking of picking one up for a trip down memory lane 😉
$0.50/1 Lip Smacker Product printable coupon (preclipped)
At Target:
BUY ONE LIPSMACKER, regular price $1.44-$1.48
Use one $0.50/1 Lip Smacker Product printable coupon (preclipped)
Extreme Couponing Tip: You can usually get two prints of most internet coupons. To find them in the easiest possible way, use our preclipped links that take you right to the coupon. Then come back and click the same link to get your 2nd print. This will save you the time spent searching for the coupon.
Never miss another hot coupon, deal or freebie again! "Like" Couponaholic .NET on Facebook, Follow us on Pinterest, Follow us on Twitter, Follow us on Instagram or Subscribe to our Daily Email Newsletter!


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|David Haraburda <david-coc...@haraburda.com>
|Re: [help needed] Patching the Servlet API on classloading
|Tue, 18 Jun 2002 19:47:16 GMT
Sylvain Wallez wrote: > > . . . > Another related item : while updating the ParanoidClassloader to make it > really paranoid, I encountered an inconsistency in java.lang.ClassLoader > class : getResource() can be overriden, but getResources() is declared > final, meaning there is no way to change the order in which e.g. > indentical service definitions in META-INF/services are enumerated. > Although this shouldn't cause a problem with JAXP (at least the Apache > implementation) which uses getResource() (no enumeration), we should be > aware that this is a potential pitfall. This isn't an inconsistency, it's intentional -- getResources() is a final method that calls findResources, a protected method on the current instance of the ClassLoader. ClassLoader implementations should override findResources(). The purpose of getResources() is to get a set of resources from all the classloaders in the classloader heirarchy. > > > As this item impacts one of the core classes of the JDK, it may be > difficult to convice people to fix it, but this would allow to uniformly > override the search order in specialized classloaders. > > Sylvain David --------------------------------------------------------------------- To unsubscribe, e-mail: email@example.com For additional commands, email: email@example.com
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Brian Stepanek is an American actor who has been in eighty millions commercials, including the "flamethrower" one for Dairy Queen, as well as being the face of the "Dex" phone book company (for you Rocky Mountain region livin' folks). He's also been in several TV shows and movies, notably having a part in Michael Bay's pre-Transformers movie The Island, and is Arwin on the Disney Channel's The Suite Life of Zack and Cody.
It's RAINING BEER!
Transformers (2007)
- "Sector Seven Agent #1" (In the SUV with Agent Simmons, "This is such a felony, what you're doing.")
Transformers: Beginnings
- Agent Simmons
- Blackout