is any point attaining . In general, a convex optimization problem may have zero, one, or many solutions. Many optimization problems can be equivalently formulated in this standard form. For example, the problem of maximizing a concave function can be re-formulated equivalently as the problem of minimizing the convex function . The problem of maximizing a concave function over a convex set is commonly called a convex optimization problem. Properties The following are useful properties of convex optimization problems: every local minimum is a global minimum; the optimal set is convex; if the objective function is strictly convex, then the problem has at most one optimal point. These results are used by the theory of convex minimization along with geometric notions from functional analysis (in Hilbert spaces) such as the Hilbert projection theorem, the separating hyperplane theorem, and Farkas' lemma. Applications The following problem classes are all convex optimization problems, or can be reduced to convex optimization problems via simple transformations: Least squares Linear programming Convex quadratic minimization with linear constraints Quadratic minimization with convex quadratic constraints Conic optimization Geometric programming Second order cone programming Semidefinite programming Entropy maximization with appropriate constraints Convex optimization has practical applications for the following. Portfolio optimization. Worst-case risk analysis. Optimal advertising. Variations of statistical regression (including regularization and quantile regression). Model fitting (particularly multiclass classification). Electricity generation optimization. Combinatorial optimization. Non-probabilistic modelling of uncertainty. Localization using wireless signals Lagrange multipliers Consider a convex minimization problem given in standard form by a cost function and inequality constraints for . Then the domain is: The Lagrangian function for the problem is For each point in that minimizes over , there exist real numbers called Lagrange multipliers, that satisfy these conditions simultaneously: minimizes over all with at least one (complementary slackness). If there exists a "strictly feasible point", that is, a point satisfying then the statement above can be strengthened to require that . Conversely, if some in satisfies (1)–(3) for scalars with then is certain to minimize over . Algorithms Unconstrained convex optimization can be easily solved with gradient descent (a special case of steepest descent) or Newton's method, combined with line search for an appropriate step size; these can be mathematically proven to converge quickly, especially the latter method. Convex optimization with linear equality constraints can also be solved using KKT matrix techniques if the objective function is a quadratic function (which generalizes to a