Abstract
In this paper, we present a sequential semidefinite programming (SSDP) algorithm for nonlinear semidefinite programming. At each iteration, a linear semidefinite programming subproblem and a modified quadratic semidefinite programming subproblem are solved to generate a master search direction. In order to avoid Maratos effect, a second-order correction direction is determined by solving a new quadratic programming. And then a penalty function is used as a merit function for arc search. The superlinear convergence is shown under the strict complementarity and the strong second-order sufficient conditions with the sigma term. Finally, some preliminary numerical results are reported.
Highlights
Consider the following nonlinear semidefinite programming (NLSDP) with a negative semidefinite matrix constraint: min f (x) (1.1)s.t
There are a lot of literature for NLSDP on algorithms, for example, the augmented Lagrangian method [7,8,9,10,11,12], primal-dual interior point method [13, 14], and sequential semidefinite programming (SSDP) method [15,16,17,18,19,20,21]
A penalty function is used as a merit function for arc search
Summary
Consider the following nonlinear semidefinite programming (NLSDP) with a negative semidefinite matrix constraint: min f (x) (1.1). At each iteration of SSDP method, a special quadratic semidefinite programming subproblem is solved to generate a search direction. The optimal solution dk to QSDP(xk, Hk) (2.8) cannot be guaranteed to avoid the Maratos effect and get superlinear convergence, so it needs a modification. To this end, motivated by the ideas in [20], we introduce a second-order correction direction by solving the following subproblem: min dk + d dk + d THk dk + d s.t. NkT A xk + dk + DA xk d Nk = – dk Em–r,. Lemma 2.6 Suppose that Assumptions A1–A2 hold, dk is the optimal solution of QSDP(xk, Hk) (2.8).
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