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semismooth Newton methods for solving semi-infinite programming problems

JOURNAL OF GLOBAL OPTIMIZATION 2003年第2-3期27卷 215-232页

作者： Qi, LQ Wu, SY Zhou, GL Hong Kong Polytech Univ Dept Math Appl Kowloon Hong Kong Peoples R China Natl Cheng Kung Univ Inst Appl Math Tainan 700 Taiwan Natl Univ Singapore Dept Math Singapore 117548 Singapore

In this paper we present some semismooth Newton methods for solving the semi-infinite programming problem. We first reformulate the equations and nonlinear complementarity conditions derived from the problem into a system of semismooth equations by using NCP functions. Under some conditions a solution of the system of semismooth equations is a solution of the problem. Then some semismooth Newton methods are proposed for solving this system of semismooth equations. These methods are globally and superlinearly convergent. Numerical results are also given.

关键词： semi-infinite programming semismooth equations semismooth Newton method

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A quasi-Newton interior point method for semi-infinite programming

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OPTIMIZATION METHODS & SOFTWARE 2003年第6期18卷 673-687页

作者： Vaz, AIF Fernandes, EMGP Gomes, MPSF Univ Minho Escola Engn Dept Prod & Sistemas P-4710057 Braga Portugal Univ London Imperial Coll Sci Technol & Med Dept Mech Engn Mechatron Med Lab London SW7 2BX England

We propose an interior point method for solving nonlinear semi-infinite programming (SIP) problems. The method is based on a transcription of the SIP problem into a nonlinear finite problem. We apply the interior point strategy to the finite problem and obtain an algorithm for nonlinear SIP. The evaluation of the constraint functions and derivatives of the finite problem requires the numerical evaluation of integrals that we compute by an adaptative trapezoid formula. Numerical results with a collection of several problems from the SIPAMPL [A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes. SIPAMPL v2.0: semi-infinite programming with AMPL. Technical Report ALG/EF/4-2002, Universidade do Minho, Braga, Portugal, December 2002. [http://***/aivaz/].] database are also shown.

关键词： quasi-Newton method interior point method semi-infinite programming

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A quasi-Newton interior point method for semi-infinite programming

A quasi-Newton interior point method for semi-infinite progr...

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1st International Conference on Optimization Methods and Software (OMS2002)

关键词： quasi-Newton method interior point method semi-infinite programming

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semismooth Newton methods for solving semi-infinite programming problems

Semismooth Newton methods for solving semi-infinite programm...

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International Workshop on Optimization with High Technology Applications

关键词： semi-infinite programming semismooth equations semismooth Newton method

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Ill-posedness with respect to the solvability in linear optimization

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LINEAR ALGEBRA AND ITS APPLICATIONS 2006年第2-3期416卷 520-540页

作者： Canovas, M. J. Lopez, M. A. Parra, J. Toledo, F. J. Univ Alicante Dept Stat & Operat Res Alicante 03071 Spain Miguel Hernandez Univ Elche Ctr Operat Res Alicante 03202 Spain

We characterize those linear optimization problems that are ill-posed in the sense that arbitrarily small perturbations of the problem's data may yield both, solvable and unsolvable problems. Thus, the ill-posedness is identified with the boundary of the set of solvable problems. The associated concept of well-posedness turns out to be equivalent to different stability criteria traced out from the literature of linear programming. Our results, established for linear problems with arbitrarily many constraints, also provide a new insight for the ill-posedness in ordinary and conic linear programming. They are formulated in terms of suitable subsets of R-n and Rn+1 (n is the number of unknowns) which only depend on the problem coefficients. (c) 2006 Elsevier Inc. All rights reserved.

关键词： linear optimization III-posedness stability semi-infinite programming

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SECOND-ORDER NONSMOOTH OPTIMIZATION FOR H SYNTHESIS

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IFAC Proceedings Volumes 2006年第9期39卷 501-506页

作者： Vincent Bompart Dominikus Noll Pierre Apkarian ONERA-CERT 2 av. Edouard Belin 31055 Toulouse France and Université Paul Sabatier Institut de Mathématiques 118 route de Narbonne 31062 Toulouse France UPS-Institut de Mathématiques ONERA-CERT and UPS-Institut de Mathématiques

We consider optimization programs which arise in automatic control applications for H ∞ controller synthesis. We optimize functions which are finite or infinite maxima of smooth functions, or of semismooth functions like the maximum eigenvalue functions. From a nonsmooth semi-infinite problem formulation, a second-order algorithm is developed. Our method is tested on several examples in controller synthesis.

关键词： H ∞ synthesis semi-infinite programming sequential quadratic programming (SQP) nonsmooth optimization trust-region

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On Nesterov's approach to semi-infinite programming

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ACTA APPLICANDAE MATHEMATICAE 2002年第2期74卷 195-215页

作者： Faybusovich, L Univ Notre Dame Dept Math Notre Dame IN 46556 USA

We generalize Nesterov's construction for the reduction of various classes of semi-infinite programming problems to the semidefinite programming form. In this way, we are able to consider 'cones of squares' of real-valued and matrix-valued functions as rather particular cases of a unifying abstract scheme. We also interpret from this viewpoint some results of M. Krein and A. Nudelman. This provides (in a way which probably has not been anticipated by these authors) a very powerful tool for solving various optimization problems.

关键词： interior-point algorithms moment problems semi-infinite programming semidefinite programming nonnegative polynomials

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An adaptive dual parametrization algorithm for quadratic semi-infinite programming problems

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JOURNAL OF GLOBAL OPTIMIZATION 2002年第2期24卷 205-217页

作者： Liu, Y Teo, KL Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China

The so called dual parametrization method for quadratic semi-infinite programming (SIP) problems is developed recently for quadratic SIP problems with a single infinite constraint. A dual parametrization algorithm is also proposed for numerical solution of such problems. In this paper, we consider quadratic SIP problems with positive definite objective and multiple linear infinite constraints. All the infinite constraints are supposed to be continuously dependent on their index variable on a compact set which is defined by a number equality and inequalities. We prove that in the multiple infinite constraint case, the minimu parametrization number, just as in the single infinite constraint case, is less or equal to the dimension of the SIP problem. Furthermore, we propose an adaptive dual parametrization algorithm with convergence result. Compared with the previous dual parametrization algorithm, the adaptive algorithm solves subproblems with much smaller number of constraints. The efficiency of the new algorithm is shown by solving a number of numerical examples.

关键词： semi-infinite programming global optimization dual parametrization adaptive algorithm convergence

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Optimization of metabolic pathways under stability considerations

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COMPUTERS & CHEMICAL ENGINEERING 2005年第3期29卷 467-479页

作者： Chang, YJ Sahinidis, NV Univ Illinois Dept Chem & Biomol Engn Urbana IL 61801 USA

Metabolic networks are often approached through steady-state optimization formulations that are solved to interpret and predict the behavior of the network subject to changes in external fluxes or internal enzyme activity. The major question addressed in this paper is how to ensure that solutions to these steady-state optimization models for metabolic networks are implementable from a stability point of view. The stability of a dynamic system is closely related to matrix stability. Hence, it can be determined through the computation of the largest eigenvalue of a coefficient matrix. While it is straightforward to analyze the stability of a given system, the challenge is to redesign a metabolic network in a way that guarantees that the system will be stable around the new steady-state. For this purpose, we propose to model metabolic networks through classical optimization formulations, such as the classical S-system representation, with an additional constraint to enforce stability within a prespecified neighborhood of the solution point. The proposed formulation is a bilevel optimization problem that is very difficult to solve. We develop a suitable global optimization algorithm to solve this problem after transforming it to a semi-infinite optimization problem. Computational results are presented, including application to tryptophan biosynthesis in bacteria and anaerobic fermentation in Saccharomyces cerevisiae. (c) 2004 Elsevier Ltd. All rights reserved.

关键词： metabolic engineering dynamic stability semi-infinite programming

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Metric regularity of semi-infinite constraint systems

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MATHEMATICAL programming 2005年第2-3期104卷 329-346页

作者： Cánovas, MJ Dontchev, L López, MA Parra, J Miguel Hernandez Univ Elche Ctr Operat Res Alicante 03202 Spain Math Reviews Ann Arbor MI 48107 USA Univ Alicante Dept Stat & Operat Res Alicante 03071 Spain

We obtain a formula for the modulus of metric regularity of a mapping defined by a semi-infinite system of equalities and inequalities. Based on this formula, we prove a theorem of Eckart-Young type for such set-valued infinite-dimensional mappings: given a metrically regular mapping F of this kind, the infimum of the norm of a linear function g such that F+g is not metrically regular is equal to the reciprocal to the modulus of regularity of F. The Lyusternik-Graves theorem gives a straightforward extension of these results to nonlinear systems. We also discuss the distance to infeasibility for homogeneous semi-infinite linear inequality systems.

关键词： semi-infinite programming metric regularity distance to inconsistency conditioning

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