Primal-dual path following method for nonlinear semi-infinite programs with semi-definite constraints

In this paper, we propose a primal-dual path following method for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints, called SISDP for short. A straightforward approach to the SISDP is to use classical methods for semi-infinite programs such as discreti...

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Veröffentlicht in:Mathematical programming Jg. 199; H. 1-2; S. 251 - 303
Hauptverfasser: Okuno, Takayuki, Fukushima, Masao
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2023
Springer
Schlagworte:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Methods
Numerical Analysis
Theoretical
Global convergence
Semi-infinite program
Nonlinear semi-definite program
90C22
Path-following method
90C34
Superlinear convergence
90C26
ISSN:0025-5610, 1436-4646
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Zusammenfassung:In this paper, we propose a primal-dual path following method for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints, called SISDP for short. A straightforward approach to the SISDP is to use classical methods for semi-infinite programs such as discretization and exchange methods and solve a sequence of (nonlinear) semi-definite programs (SDPs). However, it is often too demanding to find exact solutions of SDPs. In contrast, our approach does not rely on solving SDPs accurately but approximately following a path leading to a solution, which is formed on the intersection of the semi-infinite feasible region and the interior of the semi-definite feasible region. Specifically, we first present a prototype path-following method and show its global weak* convergence to a Karush-Kuhn-Tucker point of the SISDP under some mild assumptions. Next, to achieve fast local convergence, we integrate a two-step sequential quadratic programming method equipped with the Monteiro-Zhang scaling technique into the prototype method. We prove two-step superlinear convergence of the resulting algorithm using Alizadeh-Hareberly-Overton-like, Nesterov-Todd, and Helmberg-Rendle-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro scaling directions. Finally, we conduct some numerical experiments to demonstrate the efficiency of the proposed method through comparison with a discretization method that solves SDPs obtained by finite relaxation of the SISDP.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-022-01827-2