# The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method

Save this PDF as:

Size: px
Start display at page:

Download "The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method"

## Transcription

1 The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method Robert M. Freund February, Massachusetts Institute of Technology. 1

2 1 The Algorithm The problem we are interested in solving is: P : minimize f(x) s.t. x R n, where f(x) is differentiable. If x = x is a given point, f(x) can be approximated by its linear expansion f( x + d) f( x)+ f( x) T d if d small, i.e., if d is small. Now notice that if the approximation in the above expression is good, then we want to choose d so that the inner product f( x) T d is as small as possible. Let us normalize d so that d = 1. Then among all directions d with norm d = 1, the direction d = f( x) f( x) makes the smallest inner product with the gradient f( x). This fact follows from the following inequalities: ( ) f( x) d = f( ) T f( x) T x) d f( x = f( ) x T d. f( x) For this reason the un-normalized direction: d = f( x) is called the direction of steepest descent at the point x. Note that d = f( x) is a descent direction as long as f( x) =0. To see this, simply observe that d T f( x) = ( f( x)) T f( x) < 0solongas f( x) =0.

3 A natural consequence of this is the following algorithm, called the steepest descent algorithm. Steepest Descent Algorithm: Step 0. Given x 0,set k := 0 Step 1. d k := f (x k ). If d k = 0, then stop. Step. Solve min α f (x k + αd k ) for the stepsize α k, perhaps chosen by an exact or inexact linesearch. Step 3. Set x k+1 x k + α k d k,k k +1. Go to Step 1. Note from Step and the fact that d k = f (x k ) is a descent direction, it follows that f (x k+1 ) < f (x k ). Global Convergence We have the following theorem: Convergence Theorem: Suppose that f ( ) : R n R is continuously differentiable on the set S = {x R n f (x) f (x 0 )}, and that S is a closed and bounded set. Then every point x that is a cluster point of the sequence {x k } satisfies f ( x) =0. Proof: The proof of this theorem is by contradiction. By the Weierstrass Theorem, at least one cluster point of the sequence {x k } must exist. Let x be any such cluster point. Without loss of generality, assume that lim k k x = x, but that f ( x) = 0. This being the case, there is a value of α> 0 such that δ := f ( x) f ( x + αd) > 0, where d = f ( x). Then also ( x + αd) ints. 3

4 Now lim k d k = d. Then since ( x + αd) ints, and (x k + αd k ) ( x + αd), for k sufficiently large we have However, δ f (x k αd k δ + ) f ( x + αd)+ x) δ + δ = f ( = f ( x). f ( x) < f (x k + α k d k ) f (x k + αd k ) f ( x) which is of course a contradiction. Thus d = f ( x) =0. q.e.d. δ, 3 The Rate of Convergence for the Case of a Quadratic Function In this section we explore answers to the question of how fast the steepest descent algorithm converges. We say that an algorithm exhibits linear convergence in the objective function values if there is a constant δ< 1such that for all k sufficiently large, the iterates x k satisfy: f (x k+1 ) f (x ) f (x k ) f (x δ, ) where x is some optimal value of the problem P. The above statement says that the optimality gap shrinks by at least δ at each iteration. Notice that if δ = 0.1, for example, then the iterates gain an extra digit of accuracy in the optimal objective function value at each iteration. If δ = 0.9, for example, then the iterates gain an extra digit of accuracy in the optimal objective function value every iterations, since (0.9) The quantity δ above is called the convergence constant. We would like this constant to be smaller rather than larger. We will show now that the steepest descent algorithm exhibits linear convergence, but that the convergence constant depends very much on the ratio of the largest to the smallest eigenvalue of the Hessian matrix H(x) at 4

5 the optimal solution x = x. In order to see how this arises, we will examine the case where the objective function f (x) is itself a simple quadratic function of the form: f (x) = 1 x T Qx + q T x, where Q is a positive definite symmetric matrix. We will suppose that the eigenvalues of Q are A = a 1 a... a n = a > 0, i.e, A and a are the largest and smallest eigenvalues of Q. The optimal solution of P is easily computed as: x = Q 1 q and direct substitution shows that the optimal objective function value is: 1 T Q 1 f (x )= q q. For convenience, let x denote the current point in the steepest descent algorithm. We have: f (x) = 1 x T Qx + q T x and let d denote the current direction, which is the negative of the gradient, i.e., d = f (x) = Qx q. Now let us compute the next iterate of the steepest descent algorithm. If α is the generic step-length, then 1 f (x + αd) = (x + αd) T Q(x + αd)+ q T (x + αd) 5

6 1 T = 1 x T Qx + αd T Qx + α d T Qd + q x + αq T d 1 = f (x) αd T d + α d T Qd. Optimizing the value of α in this last expression yields d T d α = d T Qd, and the next iterate of the algorithm then is and d T d x = x + αd = x + d, d T Qd 1 (d T d) f (x )= f (x + αd) = f (x) αd T d + 1 α d T Qd = f (x). d T Qd Therefore, f (x ) f (x ) f (x) 1 (d T d) f (x ) d = T Qd f (x) f (x ) f (x) f (x ) =1 =1 1 (d T d) d T Qd 1 x T Qx + q T x + 1 qt Q 1 q 1 (d T d) d T Qd 1 (Qx + q)t Q 1 (Qx + q) (d T d) =1 (d T Qd)(d T Q 1 d) 1 =1 β where (d T Qd)(d T Q 1 d) β =. (d T d) 6

7 In order for the convergence constant to be good, which will translate to fast linear convergence, we would like the quantity β to be small. The following result provides an upper bound on the value of β. Kantorovich Inequality: Let A and a be the largest and the smallest eigenvalues of Q, respectively. Then β (A + a). 4Aa We will prove this inequality later. For now, let us apply this inequality to the above analysis. Continuing, we have ( f (x ) f (x ) 1 4Aa = (A a) A/a 1 ) =1 1 = =: δ. f (x) f (x ) β (A + a) (A + a) A/a +1 Note by definition that A/a is always at least 1. If A/a is small (not much bigger than 1), then the convergence constant δ will be much smaller than 1. However, if A/a is large, then the convergence constant δ will be only slightly smaller than 1. Table 1 shows some sample values. Note that the number of iterations needed to reduce the optimality gap by a factor of 10 grows linearly in the ratio A/a. 4 Examples 4.1 Example 1: Typical Behavior In step 1 of the steepest descent algorithm, we need to compute ( ) ( ) k k k 10x1 4xk +14 dk 1 d k = f (x1,x )= xk = 4xk 1 +6 dk 7 Consider the function f (x 1,x )=5x 1 + x +4x 1 x 14x 1 6x + 0. This function has its optimal solution at x =(x 1,x )=(1, 1) and f (1, 1) = 10.

8 A a Upper Bound on δ = ( A/a 1 A/a+1 ) Number of Iterations to Reduce the Optimality Gap by a factor of Table 1: Sensitivity of Steepest Descent Convergence Rate to the Eigenvalue Ratio andinstepwe need to solve α k =arg min α h(α) = f (x k + αd k ). In this example we will be able to derive an analytic expression for α k. Notice that h(α) = f (x k + αd k ) = k k k k 5(x 1 + αd k 1) +(x ) +4(x 1 + αd k + αd k 1)(x + αd) k k k 14(x 1) 6(x + αd k 1 + αd k )+0, and this is a simple quadratic function of the scalar α. It is minimized at α k = (d k 1 ) +(d k ) (5(d k ) +4d k ) 1 ) +(d k 1 dk Using the steepest descent algorithm to minimize f (x) starting from 1 1 x 1 =(x 1,x )=(0, 10), and using a tolerance of ɛ = 10 6, we compute the iterates shown in Table and in Figure : For a convex quadratic function f (x) = 1 x T Qx c T x, the contours of the function values will be shaped like ellipsoids, and the gradient vector f (x) at any point x will be perpendicular to the contour line passing through x, see Figure 1. 8

9 k x k 1 x k k d 1 k d d k α k f (x k ) Table : Iterations of the example in Subsection

10 5 x +4 y +3 x y+7 x y x Figure 1: The contours of a convex quadratic function are ellipsoids. 4. Example : Hem-Stitching Consider the function f(x) = 1 xt Qx c T x +10 where Q and c are given by: and Q = c = ( ( 14 6 ) ) 10

11 Figure : Behavior of the steepest descent algorithm for the example of Subsection 4.1. A For this problem, a =30.34, and so the linear convergence rate δ of ( A/a 1 ) the steepest descent algorithm is bounded above by δ = A/a+1 = If we apply the steepest descent algorithm to minimize f (x) starting from x 1 =(40, 100), we obtain the iteration values shown in Table 3 and shown graphically in Figure Example 3: Rapid Convergence Consider the function f (x) = 1 x T Qx c T x

12 k x k 1 x k d k α k f (x k ) f (x k ) f (x ) f (x k 1 ) f (x ) Table 3: Iteration values for example of Subsection 4., which shows hemstitching. where Q and c are given by: ( ) 0 5 Q = 5 16 and ( ) 14 c = 6 A For this problem, a =1.8541, and so the linear convergence rate δ of ( ) A 1 a the steepest descent algorithm is bounded above by δ = = A +1 a 1

13 Figure 3: Hem-stitching in the example of Subsection 4.. If we apply the steepest descent algorithm to minimize f (x) starting from x 1 =(40, 100), we obtain the iteration values shown in Table 4 and shown graphically in Figure 4. Figure 4: Rapid convergence in the example of Subsection

14 k x k 1 x k d k α k f (x k ) f (x k ) f (x ) f (x k 1 ) f (x ) Table 4: Iteration values for the example in Subsection 4.3, which exhibits rapid convergence. 4.4 Example 4: A nonquadratic function Let 4 4 f (x) = x 1 0.6x +4x x 4 log(x i ) log(5 x i ). i=1 i=1 Table 5 shows values of the iterates of steepest descent applied to this function. 4.5 Example 5: An analytic example Suppose that f (x) = 1 x T Qx + q T x 14

15 k x k 1 x k x k 3 x k 4 d k f (x k ) f (x k ) f (x ) f (x k 1 ) f (x ) Table 5: Iteration values for the example of Subsection

16 where ( ) ( ) +4 + Q = and q =. + Then ( )( ) ( ) +4 x + f (x) = x and so ( ) 0 x = 1 and f (x )= 1. Direct computation shows that the eigenvalues of Q are A =3 + 5and a = 3 5, whereby the bound on the convergence constant is ( A/a 1 ) δ = A/a +1 Suppose that x 0 =(0, 0). Then we have: x 1 =( 0.4, 0.4), x =(0, 0.8) and the even numbered iterates satisfy x n =(0, 1 0. n ) and f (x n )=(1 0. n ) + (0.) n and so f (x n ) f (x )=(0.) n. Therefore, starting from the point x 0 =(0, 0), the optimality gap goes down by a factor of 0.04 after every two iterations of the algorithm. 5 Final Remarks The proof of convergence is basic in that it only relies on fundamental properties of the continuity of the gradient and on the closedness and boundedness of the level sets of the function f (x). The analysis of the rate of convergence is quite elegant. 16

17 Kantorovich won the Nobel Prize in Economics in 1975 for his contributions to the application of linear programming to economic theory. The bound on the rate of convergence is attained in practice quite often, which is too bad. The ratio of the largest to the smallest eigenvalue of a matrix is called the condition number of the matrix. The condition number plays an extremely important role in the theory and the computation of quantities involving the matrix Q. What about non-quadratic functions? Most functions behave as nearquadratic functions in a neighborhood of the optimal solution. The analysis of the non-quadratic case gets very involved; fortunately, the key intuition is obtained by analyzing the quadratic case. 6 A Bisection Algorithm for a Line-Search of a Convex Function Suppose that f(x) is a continuously differentiable convex function, and that we seek to solve: α := arg min f( x + αd ), α where x is our current iterate, and d is the current direction generated by an algorithm that seeks to minimize f(x). Suppose further that d is a descent direction of f(x) at x =, x namely: Let f( x + ɛd ) < f( x) for all ɛ >0 and sufficiently small. h(α) := f( x + αd ), whereby h(α) is a convex function in the scalar variable α, and our problem is to solve for ᾱ =arg min h(α). α We therefore seek a value ᾱ for which h (ᾱ) =0. 17

18 It is elementary to show that Proposition 6.1 h (0) < 0. q.e.d. h (α) = f ( x + αd) T d. Because h(α) is a convex function of α, wealsohave: Proposition 6. h (α) is a monotone increasing function α. of q.e.d. Figure 5 shows an example of convex function of two variables to be optimized. Figure 6 shows the function h(α) obtained by restricting the function of Figure 5 to the line shown in that figure. Note from Figure 6 that h(α) is convex. Therefore its first derivative h (α) will be a monotonically increasing function. This is shown in Figure 7. Because h (α) is a monotonically increasing function, we can approxiα, the point that satisfies h ( α) = 0, by a suitable bisection mately compute method. Suppose that we know a value α ˆ that h (ˆ α) > 0. Since h (0) < 0 and h (ˆ α) > 0, the mid-value α = 0+ α ˆ is a suitable test-point. Note the following: If h ( α) = 0, we are done. If h ( α) > 0, we can now bracket α in the interval (0, α). If h ( α) < 0, we can now bracket α in the interval ( α, α). ˆ This leads to the following bisection algorithm for minimizing h(α) = f ( x + αd) by solving the equation h (α) 0. Step 0. Set k = 0. Set α l := 0 and α u := ˆα. α = αu+α l Step k. Set and compute h ( α). 18

19 f(x,x 30) x Figure 5: A convex function to be optimized. 19 x1

20 h(α) α Figure 6: The 1-dimensional function h(α) h (α) α Figure 7: The function h (α) is monotonically increasing. 0

21 If h ( α) > 0, re-set α u := α. Set k k +1. If h ( α) < 0, re-set α l := α. Set k k +1. If h ( α) = 0, stop. Proposition 6.3 After every iteration of the bisection algorithm, the current interval [α l,α u ] must contain a point α such that h ( α) =0. Proposition 6.4 At the k th iteration of the bisection algorithm, the length of the current interval [α l,α u ] is ( ) 1 k L = (ˆα). Proposition 6.5 A value of α such that α ᾱ ɛ canbefoundinat most ( ) ˆα log ɛ steps of the bisection algorithm. 6.1 Computing α for which h (ˆ ˆ α) > 0 Suppose that we do not have available a convenient value αˆ for which h (ˆ α) > 0. One way to proceed is to pick an initial guess of α ˆ and compute h (ˆ α). If h (ˆ α) > 0, then proceed to the bisection algorithm; if h (ˆ α) 0, then re-set α ˆ ˆ α and repeat the process. 6. Stopping Criteria for the Bisection Algorithm In practice, we need to run the bisection algorithm with a stopping criterion. Some relevant stopping criteria are: Stop after a fixed number of iterations. That is stop when k = K, where K specified by the user. Stop when the interval becomes small. That is, stop when α u α l ɛ, where ɛ is specified by the user. Stop when h ( α) becomes small. That is, stop when h ( α) ɛ, where ɛ is specified by the user. This third stopping criterion typically yields the best results in practice. 1

22 6.3 Modification of the Bisection Algorithm when the Domain of f (x) is Restricted The discussion and analysis of the bisection algorithm has presumed that our optimization problem is P : minimize f (x) n s.t. x R. Given a point x and a direction d, the line-search problem then is LS : minimize h(α) := f ( x + αd ) s.t. α R. n Suppose instead that the domain of definition of f (x) isanopenset X R. Then our optimization problem is: P : minimize f (x) s.t. x X, and the line-search problem then is LS : minimize h(α) := f ( x + αd ) s.t. x + αd X. In this case, we must ensure that all iterate values of α in the bisection alx + αd X. As an example, consider the following gorithm satisfy problem: m P : minimize f (x) := ln(b i A i x) i=1 s.t. b Ax > 0.

23 Here the domain of f(x) is X = {x R b Ax > 0}. Given x X and a direction d, the line-search problem is: n a point LS : minimize h(α) := f( x + αd ) = m ln(b i A i ( x + αd )) i=1 s.t. b A( x + αd ) > 0. Standard arithmetic manipulation can be used to establish that b A( x + αd ) > 0 if and only if α<α< ˇ αˆ where { b } { } i A i x b i A i x α ˇ := min and α ˆ := min, A i d<0 A i d A i d>0 A i d and the line-search problem then is: m LS : minimize h(α) := ln(b i A i ( x + αd )) i=1 s.t. α<α< ˇ α. ˆ 3

24 7 Proof of Kantorovich Inequality Kantorovich Inequality: Let A and a be the largest and the smallest eigenvalues of Q, respectively. Then (A + a) β. 4Aa Proof: Let Q = RDR T, and then Q 1 = RD 1 R T, where R = R T is an orthonormal matrix, and the eigenvalues of Q are 0 < a = a 1 a... a n = A, and, a a... 0 D = a n Then β = (d T RDR T d)(d T RD 1 R T d) f T Dff T D 1 f = (d T RR T d)(d T RR T d) f T ff T f n where f = R T f d.let λ i = f T i f. Then λ i 0and λ i =1, and i=1 n ( ) 1 n n ( ) λ 1 i a i i=1 β = λ i a i λ i =. a i=1 i=1 i 1 n λ i a i i=1 The largest value of β is when λ 1 + λ n = 1, see the illustration in Figure 8. Therefore, 4

25 1 1 λ 1 a + λ n A = (λ 1a + λ n A)(λ 1 A + λ n a) ( 1 A + 1 a)( 1 a + 1 A) (A + a) β 1 =. Aa Aa 4Aa q.e.d. λ 1 a+λ na y y = 1/x Σλ i (1/a i ) 1/(Σλ i a i ) 0 Σλ i a i 0 a 1 a a 3... a n x =a =A Figure 8: Illustration for proof of the Kantorovich Inequality. 5

26 8 Steepest Descent Exercises NOTE ON COMPUTATION EXERCISES: You may use any machine and any programming language. We recommend, however, that you use MATLAB on Athena or on your own computer. (The teaching assistant is prepared to help you get started in MATLAB on Athena.) Some of the specific details in the computation exercises are purposely left to your own discretion. For example, in cases where you must use a line-search routine, you must choose your own tolerance for the line-search. Also, you must choose your own stopping criteria for your computation. Please include your code or pseudo-code in your write-up. 1. (Steepest Descent) Suppose that x k and x k+1 are two consecutive points generated by the steepest descent algorithm with exact linesearch. Show that f(x k ) T f(x k+1 )=0.. (Steepest Descent) Suppose that we seek to minimize f(x 1,x )=5x +5x x 1 x 11x 1 +11x (a) Find a point satisfying the first-order necessary conditions for a solution. (b) Show that this point is a global minimum of f(x). (c) What would be the worst rate of convergence for the steepest descent algorithm for this problem? (d) Starting at (x 1,x )=(0, 0), at most how many steepest descent iterations would it take to reduce the function value to 10 11? 3. (Steepest Descent) Suppose we seek to minimize 1 T f(x) = x T Hx + c x +13, where ( ) ( ) H = and c = Implement the steepest descent algorithm on this problem, using the following starting points: 6

27 x 0 =(0, 0) T. x 0 =( 0.4, 0) T. x 0 =(10, 0) T. x 0 =(11, 0) T. As it turns out, the optimal solution to this problem is x =(5, 6) T, with f(x )=. What linear convergence constants do you observe for each of the above starting points? 4. (Steepest Descent) Suppose we seek to minimize 1 T f(x) = x T Hx + c x, where H = and c = Implement the steepest descent algorithm on this problem, using the following starting points. x 0 =(0, 0, 0) T. x 0 =(15.09, 7.66, 6.56) T. x 0 =(11.77, 6.4, 4.8) T. x 0 =(4.46,.5, 1.85) T. As it turns out, the optimal solution to this problem is x =(4, 3, 1) T, with f(x )= What linear convergence constants do you observe for each of the above starting points? 5. (Steepest Descent) Suppose that we seek to minimize the following function: f(x 1,x )= 9x 1 10x +θ( ln(100 x 1 x ) ln(x 1 ) ln(x ) ln(50 x 1 +x )), where θ is a given parameter. Note that the domain of this function is X = {(x 1,x ) x 1 > 0,x > 0,x 1 + x < 100,x 1 x < 50}. Implement the steepest descent algorithm for this problem, using the 7

28 bisection algorithm for your line-search, with the stopping criterion that h ( α) ɛ = Run your algorithm for θ = 10 and for θ = 100, using the following starting points. x 0 =(8, 90) T. x 0 =(1, 40) T. x 0 =(15, 68.69) T. x 0 =(10, 0) T. What linear convergence constants do you observe for each of the above starting points? Helpful Hint: Note that f(x 1,x ) is only defined on the domain X = {(x 1,x ) x 1 > 0,x > 0,x 1 + x < 100,x 1 x < 50}. If you are at a point x = ( x 1, x ) and you have computed a direction d = (d 1, d ), then note that a value of the upper bound ˆα is effectively given by the largest value of α for which the following constraints are satisfied: x 1 +αd 1 > 0, x +αd > 0, x 1 +αd 1+ x +αd < 100, x 1 +αd 1 x αd < 50. The largest value of α satisfying these conditions can easily be computed by applying appropriate logic. Important Note: Please keep the value of θ as a specific command in your code. The reason for this is that later in the semester, we will consider an algorithm where we iteratively change the value of θ. It will then be useful for your code to have a line in it where θ can be set and re-set. 8

### t := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).

1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction

### Numerisches Rechnen. (für Informatiker) M. Grepl J. Berger & J.T. Frings. Institut für Geometrie und Praktische Mathematik RWTH Aachen

(für Informatiker) M. Grepl J. Berger & J.T. Frings Institut für Geometrie und Praktische Mathematik RWTH Aachen Wintersemester 2010/11 Problem Statement Unconstrained Optimality Conditions Constrained

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### Nonlinear Algebraic Equations. Lectures INF2320 p. 1/88

Nonlinear Algebraic Equations Lectures INF2320 p. 1/88 Lectures INF2320 p. 2/88 Nonlinear algebraic equations When solving the system u (t) = g(u), u(0) = u 0, (1) with an implicit Euler scheme we have

Quadratic Functions, Optimization, and Quadratic Forms Robert M. Freund February, 2004 2004 Massachusetts Institute of echnology. 1 2 1 Quadratic Optimization A quadratic optimization problem is an optimization

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### Practice with Proofs

Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

### Introduction to Convex Optimization for Machine Learning

Introduction to Convex Optimization for Machine Learning John Duchi University of California, Berkeley Practical Machine Learning, Fall 2009 Duchi (UC Berkeley) Convex Optimization for Machine Learning

### (Quasi-)Newton methods

(Quasi-)Newton methods 1 Introduction 1.1 Newton method Newton method is a method to find the zeros of a differentiable non-linear function g, x such that g(x) = 0, where g : R n R n. Given a starting

### Nonlinear Optimization: Algorithms 3: Interior-point methods

Nonlinear Optimization: Algorithms 3: Interior-point methods INSEAD, Spring 2006 Jean-Philippe Vert Ecole des Mines de Paris Jean-Philippe.Vert@mines.org Nonlinear optimization c 2006 Jean-Philippe Vert,

### Derivative Free Optimization

Department of Mathematics Derivative Free Optimization M.J.D. Powell LiTH-MAT-R--2014/02--SE Department of Mathematics Linköping University S-581 83 Linköping, Sweden. Three lectures 1 on Derivative Free

### Section 4.4 Inner Product Spaces

Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### since by using a computer we are limited to the use of elementary arithmetic operations

> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations

### x a x 2 (1 + x 2 ) n.

Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

### Date: April 12, 2001. Contents

2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........

### Numerical methods for finding the roots of a function

Numerical methods for finding the roots of a function The roots of a function f (x) are defined as the values for which the value of the function becomes equal to zero. So, finding the roots of f (x) means

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### CSCI567 Machine Learning (Fall 2014)

CSCI567 Machine Learning (Fall 2014) Drs. Sha & Liu {feisha,yanliu.cs}@usc.edu September 22, 2014 Drs. Sha & Liu ({feisha,yanliu.cs}@usc.edu) CSCI567 Machine Learning (Fall 2014) September 22, 2014 1 /

### Eigenvalues and eigenvectors of a matrix

Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a non-zero column vector V such that AV = λv then λ is called an eigenvalue of A and V is

### Nonlinear Programming Methods.S2 Quadratic Programming

Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective

### ROBERTO BATTITI, MAURO BRUNATO. The LION Way: Machine Learning plus Intelligent Optimization. LIONlab, University of Trento, Italy, Apr 2015

ROBERTO BATTITI, MAURO BRUNATO. The LION Way: Machine Learning plus Intelligent Optimization. LIONlab, University of Trento, Italy, Apr 2015 http://intelligentoptimization.org/lionbook Roberto Battiti

### Lecture 3: Convex Sets and Functions

EE 227A: Convex Optimization and Applications January 24, 2012 Lecture 3: Convex Sets and Functions Lecturer: Laurent El Ghaoui Reading assignment: Chapters 2 (except 2.6) and sections 3.1, 3.2, 3.3 of

### Lectures 5-6: Taylor Series

Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

Gradient, Subgradient and how they may affect your grade(ient) David Sontag & Yoni Halpern February 7, 2016 1 Introduction The goal of this note is to give background on convex optimization and the Pegasos

### 10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES

58 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 you saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c nn c nn...

### Lecture 5 Principal Minors and the Hessian

Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and

### Review for Calculus Rational Functions, Logarithms & Exponentials

Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

### 0 ( x) 2 = ( x)( x) = (( 1)x)(( 1)x) = ((( 1)x))( 1))x = ((( 1)(x( 1)))x = ((( 1)( 1))x)x = (1x)x = xx = x 2.

SOLUTION SET FOR THE HOMEWORK PROBLEMS Page 5. Problem 8. Prove that if x and y are real numbers, then xy x + y. Proof. First we prove that if x is a real number, then x 0. The product of two positive

### Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725

Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T

### Chapter 6. Cuboids. and. vol(conv(p ))

Chapter 6 Cuboids We have already seen that we can efficiently find the bounding box Q(P ) and an arbitrarily good approximation to the smallest enclosing ball B(P ) of a set P R d. Unfortunately, both

### ME128 Computer-Aided Mechanical Design Course Notes Introduction to Design Optimization

ME128 Computer-ided Mechanical Design Course Notes Introduction to Design Optimization 2. OPTIMIZTION Design optimization is rooted as a basic problem for design engineers. It is, of course, a rare situation

### LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12,

LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12, 2000 45 4 Iterative methods 4.1 What a two year old child can do Suppose we want to find a number x such that cos x = x (in radians). This is a nonlinear

Vector Derivatives, Gradients, and Generalized Gradient Descent Algorithms ECE 275A Statistical Parameter Estimation Ken Kreutz-Delgado ECE Department, UC San Diego November 1, 2013 Ken Kreutz-Delgado

Elementary Gradient-Based Parameter Estimation Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA Department of Music, Stanford University, Stanford, California 94305 USA Abstract

### Lecture 3: Solving Equations Using Fixed Point Iterations

cs41: introduction to numerical analysis 09/14/10 Lecture 3: Solving Equations Using Fixed Point Iterations Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mark Cowlishaw, Nathanael Fillmore Our problem,

### Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series

1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,

### Maximum and Minimum Values

Jim Lambers MAT 280 Spring Semester 2009-10 Lecture 8 Notes These notes correspond to Section 11.7 in Stewart and Section 3.3 in Marsden and Tromba. Maximum and Minimum Values In single-variable calculus,

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### The Power Method for Eigenvalues and Eigenvectors

Numerical Analysis Massoud Malek The Power Method for Eigenvalues and Eigenvectors The spectrum of a square matrix A, denoted by σ(a) is the set of all eigenvalues of A. The spectral radius of A, denoted

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### Stochastic Inventory Control

Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the

### Gradient Methods. Rafael E. Banchs

Gradient Methods Rafael E. Banchs INTRODUCTION This report discuss one class of the local search algorithms to be used in the inverse modeling of the time harmonic field electric logging problem, the Gradient

### GenOpt (R) Generic Optimization Program User Manual Version 3.0.0β1

(R) User Manual Environmental Energy Technologies Division Berkeley, CA 94720 http://simulationresearch.lbl.gov Michael Wetter MWetter@lbl.gov February 20, 2009 Notice: This work was supported by the U.S.

### Markov Chains, part I

Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X

### Lecture 13 Linear quadratic Lyapunov theory

EE363 Winter 28-9 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discrete-time

### ARE211, Fall2012. Contents. 2. Linear Algebra Preliminary: Level Sets, upper and lower contour sets and Gradient vectors 1

ARE11, Fall1 LINALGEBRA1: THU, SEP 13, 1 PRINTED: SEPTEMBER 19, 1 (LEC# 7) Contents. Linear Algebra 1.1. Preliminary: Level Sets, upper and lower contour sets and Gradient vectors 1.. Vectors as arrows.

### Principles of Scientific Computing Nonlinear Equations and Optimization

Principles of Scientific Computing Nonlinear Equations and Optimization David Bindel and Jonathan Goodman last revised March 6, 2006, printed March 6, 2009 1 1 Introduction This chapter discusses two related

### 10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES

55 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 we saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c n n c n n...

### Lecture 7: Finding Lyapunov Functions 1

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1

### Shiqian Ma, SEEM 5121, Dept. of SEEM, CUHK 1. Chapter 2. Convex Optimization

Shiqian Ma, SEEM 5121, Dept. of SEEM, CUHK 1 Chapter 2 Convex Optimization Shiqian Ma, SEEM 5121, Dept. of SEEM, CUHK 2 2.1. Convex Optimization General optimization problem: min f 0 (x) s.t., f i (x)

### Nonlinear Algebraic Equations Example

Nonlinear Algebraic Equations Example Continuous Stirred Tank Reactor (CSTR). Look for steady state concentrations & temperature. s r (in) p,i (in) i In: N spieces with concentrations c, heat capacities

### 2.3 Bounds of sets of real numbers

2.3 Bounds of sets of real numbers 2.3.1 Upper bounds of a set; the least upper bound (supremum) Consider S a set of real numbers. S is called bounded above if there is a number M so that any x S is less

### Chapter 4 Sequential Quadratic Programming

Optimization I; Chapter 4 77 Chapter 4 Sequential Quadratic Programming 4.1 The Basic SQP Method 4.1.1 Introductory Definitions and Assumptions Sequential Quadratic Programming (SQP) is one of the most

### CONSTRAINED NONLINEAR PROGRAMMING

149 CONSTRAINED NONLINEAR PROGRAMMING We now turn to methods for general constrained nonlinear programming. These may be broadly classified into two categories: 1. TRANSFORMATION METHODS: In this approach

### Bindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8

Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e

### be a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that

Problem 1A. Let... X 2 X 1 be a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that i=1 X i is nonempty and connected. Since X i is closed in X, it is compact.

### Lecture 3: Linear methods for classification

Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,

### GRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics

GRA635 Mathematics Eivind Eriksen and Trond S. Gustavsen Department of Economics c Eivind Eriksen, Trond S. Gustavsen. Edition. Edition Students enrolled in the course GRA635 Mathematics for the academic

### Math 5311 Gateaux differentials and Frechet derivatives

Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.

### GI01/M055 Supervised Learning Proximal Methods

GI01/M055 Supervised Learning Proximal Methods Massimiliano Pontil (based on notes by Luca Baldassarre) (UCL) Proximal Methods 1 / 20 Today s Plan Problem setting Convex analysis concepts Proximal operators

### v 1. v n R n we have for each 1 j n that v j v n max 1 j n v j. i=1

1. Limits and Continuity It is often the case that a non-linear function of n-variables x = (x 1,..., x n ) is not really defined on all of R n. For instance f(x 1, x 2 ) = x 1x 2 is not defined when x

### CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

### Optimization of Design. Lecturer:Dung-An Wang Lecture 12

Optimization of Design Lecturer:Dung-An Wang Lecture 12 Lecture outline Reading: Ch12 of text Today s lecture 2 Constrained nonlinear programming problem Find x=(x1,..., xn), a design variable vector of

### By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### Lecture 3: Summations and Analyzing Programs with Loops

high school algebra If c is a constant (does not depend on the summation index i) then ca i = c a i and (a i + b i )= a i + b i There are some particularly important summations, which you should probably

### 1 Norms and Vector Spaces

008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### {f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

### Convex Optimization SVM s and Kernel Machines

Convex Optimization SVM s and Kernel Machines S.V.N. Vishy Vishwanathan vishy@axiom.anu.edu.au National ICT of Australia and Australian National University Thanks to Alex Smola and Stéphane Canu S.V.N.

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### Advanced Topics in Machine Learning (Part II)

Advanced Topics in Machine Learning (Part II) 3. Convexity and Optimisation February 6, 2009 Andreas Argyriou 1 Today s Plan Convex sets and functions Types of convex programs Algorithms Convex learning

### 24. The Branch and Bound Method

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

### MATHEMATICAL BACKGROUND

Chapter 1 MATHEMATICAL BACKGROUND This chapter discusses the mathematics that is necessary for the development of the theory of linear programming. We are particularly interested in the solutions of a

### Schooling, Political Participation, and the Economy. (Online Supplementary Appendix: Not for Publication)

Schooling, Political Participation, and the Economy Online Supplementary Appendix: Not for Publication) Filipe R. Campante Davin Chor July 200 Abstract In this online appendix, we present the proofs for

### Solving nonlinear equations in one variable

Chapter Solving nonlinear equations in one variable Contents.1 Bisection method............................. 7. Fixed point iteration........................... 8.3 Newton-Raphson method........................

### TOPOLOGICAL PROOFS OF THE EXTREME AND INTERMEDIATE VALUE THEOREMS. Contents

TOPOLOGICAL PROOFS OF THE EXTREME AND INTERMEDIATE VALUE THEOREMS JAMES MURPHY Abstract. In this paper, I will present some elementary definitions in Topology. In particular, I will explain topological

### (Refer Slide Time: 00:00:56 min)

Numerical Methods and Computation Prof. S.R.K. Iyengar Department of Mathematics Indian Institute of Technology, Delhi Lecture No # 3 Solution of Nonlinear Algebraic Equations (Continued) (Refer Slide

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### Lecture 1 Newton s method.

Lecture 1 Newton s method. 1 Square roots 2 Newton s method. The guts of the method. A vector version. Implementation. The existence theorem. Basins of attraction. The Babylonian algorithm for finding

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### NONLINEAR AND DYNAMIC OPTIMIZATION From Theory to Practice

NONLINEAR AND DYNAMIC OPTIMIZATION From Theory to Practice IC-32: Winter Semester 2006/2007 Benoît C. CHACHUAT Laboratoire d Automatique, École Polytechnique Fédérale de Lausanne CONTENTS 1 Nonlinear

### Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.

Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian

### Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

### Nonparametric adaptive age replacement with a one-cycle criterion

Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk

### A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

### Definition of a Linear Program

Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### Readings. D Chapter 1. Lecture 2: Constrained Optimization. Cecilia Fieler. Example: Input Demand Functions. Consumer Problem

Economics 245 January 17, 2012 : Example Readings D Chapter 1 : Example The FOCs are max p ( x 1 + x 2 ) w 1 x 1 w 2 x 2. x 1,x 2 0 p 2 x i w i = 0 for i = 1, 2. These are two equations in two unknowns,

### Lecture 12 Basic Lyapunov theory

EE363 Winter 2008-09 Lecture 12 Basic Lyapunov theory stability positive definite functions global Lyapunov stability theorems Lasalle s theorem converse Lyapunov theorems finding Lyapunov functions 12

### 1 Homogenous and Homothetic Functions

1 Homogenous and Homothetic Functions Reading: [Simon], Chapter 20, p. 483-504. 1.1 Homogenous Functions Definition 1 A real valued function f(x 1,..., x n ) is homogenous of degree k if for all t > 0

### Fixed Point Theorems

Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation

### 1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

### SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}