Relaxed cutting plane method with convexification for solving nonlinear semi-infinite programming problems

Shiu, Ting-Jang; Wu, Soon‐Yi

doi:10.1007/s10589-011-9452-9

Cited by 6 publications

(4 citation statements)

References 24 publications

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“…In the absence of the semi-definite constraint, (1.1) becomes a nonlinear semi-infinite program (SIP) with an infinite number of convex constraints. For solving nonlinear SIPs, many researchers proposed various kinds of algorithms, for example discretization based methods [20,25], local reduction based methods [5,17,18,26], Newton-type methods [10,19], smoothing projection methods [32], convexification based methods [2,23,24,29], and so on. For an overview of the SIP, see [4,7,21] and the references therein.…”

Section: Mathematics Subject Classification (2010) 90c22 • 90c26 • 90...mentioning

confidence: 99%

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

Okuno¹,

Fukushima²

2018

Preprint

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In this paper, we propose two algorithms 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.Our first approach does not rely on solving SDPs but on approximately following a path leading to a solution, which is formed on the intersection of the semi-infinite region and the interior of the semi-definite region. We show weak* convergence of this method to a Karush-Kuhn-Tucker point of the SISDP under some mild assumptions and further provide with sufficient conditions for strong convergence. Moreover, as the second method, to achieve fast local convergence, we integrate a two-step sequential quadratic programming method equipped with Monteiro-Zhang scaling technique into the first method. We particularly prove two-step superlinear convergence of the second method 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.

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Section: Mathematics Subject Classification (2010) 90c22 • 90c26 • 90...mentioning

confidence: 99%

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

Okuno¹,

Fukushima²

2018

Preprint

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“…[11,12,30] and references therein. From the recent applications, let us mention utilizing of convex relaxations in biological systems [23], convexifications in semi-infinite programming [29,31], or application of convex relaxations in scheduling of crude oil operations [22]. See also the overview paper [10].…”

Section: Convex Underestimatorsmentioning

confidence: 99%

“…The optimal value is f * = 0. The classical αBB method computes α = (29,32), and the lower bound on f * is −231.0459. The generalization of the αBB method using nondiagonal quadratic terms improves the lower bound only to −230.90.…”

Section: Computational Studiesmentioning

confidence: 99%

The Effect of Hessian Evaluations in the Global Optimization αBB Method

Hladík

2017

Modeling, Simulation and Optimization of Complex Processes HPSC 2015

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We consider convex underestimators that are used in the global optimization αBB method and its variants. The method is based by augmenting the original nonconvex function by a relaxation term that is derived from an interval enclosure of the Hessian matrix. In this paper, we discuss the advantages of symbolic computation of the Hessian matrix. Symbolic computation often allows simplifications of the resulting expressions, which in turn means less conservative underestimators. We show by examples that even a small manipulation with the symbolic expressions, which can be processed automatically by computers, can have a large effect on the quality of underestimators.

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“…There are various algorithms developed in the literature for solving SIP problems. They include discretization methods [2,3], gradient based methods [4], cutting plane methods [5,6] and exchange methods [7].…”

Section: Introductionmentioning

confidence: 99%

Optimality Analysis of a Class of Semi-infinite Programming Problems

Feng

Chen

et al. 2020

J Optim Theory Appl

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In this paper, we consider a class of semi-infinite programming problems with a parameter. As the parameter increases, we prove that the optimal values decrease monotonically. Moreover, the limit of the sequence of optimal values exists as the parameter tends to infinity. In finding the limit, we decompose the original optimization problem into a series of subproblems. By calculating the maximum optimal values to the subproblems and applying a fixed-point theorem, we prove that the obtained maximum value is exactly the limit of the sequence of optimal values under certain conditions. As a result, the limit can be obtained efficiently by solving a series of simplified subproblems. Numerical examples are provided to verify the limit obtained by the proposed method. KeywordsSemi-infinite programming • Fixed-point theorem • Filter design • Beamformer design Mathematics Subject Classification (2000) 90C34 • 47H10 • 26B10

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Relaxed cutting plane method with convexification for solving nonlinear semi-infinite programming problems

Cited by 6 publications

References 24 publications

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

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

The Effect of Hessian Evaluations in the Global Optimization αBB Method

Optimality Analysis of a Class of Semi-infinite Programming Problems

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