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Biconvex Optimization
Biconvex optimization is a generalization of convex optimization Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems ... where the objective function and the constraint set can be biconvex. There are methods that can find the global optimum of these problems. A set B \subset X\times Y is called a biconvex set on X\times Y if for every fixed y\in Y , B_y = \ is a convex set in X and for every fixed x\in X , B_x = \ is a convex set in Y . A function f(x, y): B \to \mathbb is called a biconvex function if fixing x, f_x(y) = f(x, y) is convex over Y and fixing y, f_y(x) = f(x, y) is convex over X . A common practice for solving a biconvex problem (which does not guarantee global optimality of the solution) is alternatively updating x, y by fixing one of them and solving the corresp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard. Definition Abstract form A convex optimization problem is defined by two ingredients: * The ''objective function'', which is a real-valued convex function of ''n'' variables, f :\mathcal D \subseteq \mathbb^n \to \mathbb; * The ''feasible set'', which is a convex subset C\subseteq \mathbb^n. The goal of the problem is to find some \mathbf \in C attaining :\inf \. In general, there are three options regarding the existence of a solution: * If such a point ''x''* exists, it is referred to as an ''optimal point'' or ''solution''; the set of all optimal points is called the ''optimal set''; and the problem is called ''solvable''. * If f is unbou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
