Sequential Convex Programming Methods for A ... - Semantic Scholar
Sequential Convex Programming Methods for A Class of Structured Nonlinear Programming Zhaosong Lu∗ October 5, 2013
Abstract In this paper we study a broad class of structured nonlinear programming (SNLP) problems. In particular, we first establish the first-order optimality conditions for them. Then we propose sequential convex programming (SCP) methods for solving them in which each iteration is obtained by solving a convex programming problem. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the methods is a KKT point of the SNLP problems. In addition, we propose a variant of the SCP method for SNLP in which nonmonotone scheme and “local” Lipschitz constants of the associated functions are used. A similar convergence result as mentioned above is established. Key words: Sequential convex programming, structured nonlinear programming, firstorder methods
1
Introduction
In this paper we consider a class of structured nonlinear programming problems in the form of min f (x) + p(x) − u(x) s.t. gi (x) + qi (x) − vi (x) ≤ 0, i = 1, . . . , m, (1) x ∈ X, where X ⊆ 0 are parameters. One can observe that the above h’s are monotonically increasing functions in [0, ∞). Moreover, λt − h(t) is convex Pn n in [0, ∞) (see [9]). It implies that u(y) = i=1 (λyi − h(yi )) is convex in
Abstract In this paper we study a broad class of structured nonlinear programming (SNLP) problems. In particular, we first establish the first-order optimality conditions for them. Then we propose sequential convex programming (SCP) methods for solving them in which each iteration is obtained by solving a convex programming problem. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the methods is a KKT point of the SNLP problems. In addition, we propose a variant of the SCP method for SNLP in which nonmonotone scheme and “local” Lipschitz constants of the associated functions are used. A similar convergence result as mentioned above is established. Key words: Sequential convex programming, structured nonlinear programming, firstorder methods
1
Introduction
In this paper we consider a class of structured nonlinear programming problems in the form of min f (x) + p(x) − u(x) s.t. gi (x) + qi (x) − vi (x) ≤ 0, i = 1, . . . , m, (1) x ∈ X, where X ⊆ 0 are parameters. One can observe that the above h’s are monotonically increasing functions in [0, ∞). Moreover, λt − h(t) is convex Pn n in [0, ∞) (see [9]). It implies that u(y) = i=1 (λyi − h(yi )) is convex in
