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Convex semi-infinite programming algorithms with inexact separation oracles

OPTIMIZATION LETTERS 2025年第3期19卷 437-462页

作者： Oustry, Antoine Cerulli, Martina Inst Polytech Paris LIX F-91120 Palaiseau France Ecole Ponts Marne La Vallee France Univ Salerno Dept Comp Sci I-84084 Fisciano Italy

Solving convex semi-infinite programming (SIP) problems is challenging when the separation problem, namely, the problem of finding the most violated constraint, is computationally hard. We propose to tackle this difficulty by solving the separation problem approximately, i.e., by using an inexact oracle. Our focus lies in two algorithms for SIP, namely the cutting-planes (CP) and the inner-outer approximation (IOA) algorithms. We prove the CP convergence rate to be in O(1/k), where k is the number of calls to the limited-accuracy oracle, if the objective function is strongly convex. Compared to the CP algorithm, the advantage of the IOA algorithm is the feasibility of its iterates. In the case of a semi-infinite program with a Quadratically Constrained Quadratic programming separation problem, we prove the convergence of the IOA algorithm toward an optimal solution of the SIP problem despite the oracle's inexactness.

关键词： semi-infinite programming Inexact oracle Separation problem

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Lagrange duality and saddle point criteria for semi-infinite variational programming problem with Caputo-Fabrizio fractional derivative

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2025年 1-21页

作者： Jayswal, Anurag Uniyal, Gaurav Indian Inst Technol Indian Sch Mines Dept Math & Comp Dhanbad 826004 Jharkhand India

In this paper, we deal with a semi-infinite variational programming problem (SIVP) involving the Caputo-Fabrizio (CF) fractional derivative operator. Firstly, we formulate the Lagrange dual model for (SIVP) and then by using Slater's constraint qualification (SCQ) and convexity assumption, we establish the weak and strong duality theorems between primal and dual problems. Later on, the saddle point criteria associated with the Lagrange functional of the corresponding (SIVP) is discussed. Moreover, some numerical examples have been given to support the theoretical results.

关键词： semi-infinite programming Caputo-Fabrizio fractional derivative Variational problems Optimality conditions Duality Saddle point criteria

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semi-infinite programming

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2007年第2期180卷 491-518页

作者： Lopez, Marco Still, Georg Univ Twente Dept Appl Math NL-7500 AE Enschede Netherlands Univ Alicante Dept Estadist & Invest Operat Alicante 03080 Spain

A semi-infinite programming problem is an optimization problem in which finitely many variables appear in infinitely many constraints. This model naturally arises in an abundant number of applications in different fields of mathematics, economics and engineering. The paper, which intends to make a compromise between an introduction and a survey, treats the theoretical basis, numerical methods, applications and historical background of the field. (c) 2006 Elsevier B.V. All rights reserved.

关键词： semi-infinite programming applications linear semi-infinite programs optimality conditions numerical methods

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semi-infinite programming yields optimal disturbance model for offset-free nonlinear model predictive control

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JOURNAL OF PROCESS CONTROL 2021年 101卷 35-51页

作者： Caspari, Adrian Djelassi, Hatim Mhamdi, Adel Biegler, Lorenz T. Mitsos, Alexander Rhein Westfal TH Aachen Proc Syst Engn AVT SVT D-52074 Aachen Germany JARA Energy D-52056 Aachen Germany Forschungszentrum Julich Energy Syst Engn IEK 10 D-52425 Julich Germany Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

Offset-free nonlinear model predictive control (NMPC) can eliminate the tracking offset associated with the presence of plant-model mismatch or other persistent disturbances by augmenting the plant model with disturbances and employing an observer to estimate both the states and disturbances. Despite their importance, a systematic approach for the generation of suitable disturbance models is not available. We propose an optimization-based method to generate disturbance models based on sufficient observability conditions and generalize the theory of offset-free NMPC by allowing for (i) more measured variables than controlled variables and (ii) unmeasured controlled variables. Based on the sufficient conditions, we formulate a generalized semi-infinite program, which we reformulate and solve as a simpler semi-infinite program using a discretization algorithm. The solution furnishes the optimal disturbance model, which maximizes the set of those state, manipulated variable, and disturbance realizations, for which a sufficient observability condition is satisfied. The disturbance model is generated offline and can be used online for offset-free NMPC. We apply the approach using three case studies ranging from small scale chemical reactor cases to a medium scale polymerization reactor case. The results demonstrate the validity and usefulness of the generalized theory and show that the model generation approach successfully finds suitable disturbance models for offset-free NMPC. (C) 2021 Elsevier Ltd. All rights reserved.

关键词： Offset-free nonlinear model predictive control Optimal disturbance modeling Unknown disturbances Plant model mismatch semi-infinite programming Deterministic global optimization

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A semi-infinite programming model for earliness tardiness production planning with a genetic algorithm

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 1996年第8期31卷 95-106页

作者： Wang, DW Fang, SC N CAROLINA STATE UNIV RALEIGH NC 27695 USA

To closely describe the earliness/tardiness production planning problems in the JIT environment, a nonlinear semi-infinite programming model is proposed. Due to the issues of nonconvexivity and having infinitely many constraints, instead of applying traditional optimization approaches, a specially designed genetic algorithm with mutation along the negative gradient direction is developed. The proposed algorithm is a combination of the steepest descent method with the stochastic sampling algorithm. Some numerical results are included to show its potential for industrial applications.

关键词： production planning earliness/tardiness schedule genetic algorithm semi-infinite programming

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semi-infinite programming, duality, discretization and optimality conditions

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OPTIMIZATION 2009年第2期58卷 133-161页

作者： Shapiro, Alexander Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA

The aim of this article is to give a survey of some basic theory of semi-infinite programming. In particular, we discuss various approaches to derivations of duality, discretization, and first- and second-order optimality conditions. Some of the surveyed results are well known while others seem to be less noticed in that area of research.

关键词： semi-infinite programming conjugate duality convex analysis discretization Lagrange multipliers first- and second-order optimality conditions

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semi-infinite programming USING HIGH-DEGREE POLYNOMIAL INTERPOLANTS AND semiDEFINITE programming

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SIAM JOURNAL ON OPTIMIZATION 2017年第3期27卷 1858-1879页

作者： Papp, David North Carolina State Univ Dept Math Raleigh NC 27695 USA

In a common formulation of semi-infinite programs, the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval. If this function is sufficiently closely approximable by a polynomial or a rational function, then the semi-infinite program can be reformulated as an equivalent semidefinite program, which in turn can be solved with interior point methods very efficiently to high accuracy. On the other hand, solving this semidefinite program is challenging if the polynomials involved are of high degree, due to numerical difficulties and bad scaling arising both from the polynomial approximations and from the fact that the semidefinite programming constraints coming from the sum-of-squares representation of nonnegative polynomials are badly scaled. We combine polynomial function approximation techniques and polynomial programming to overcome these numerical difficulties, using sum-of-squares interpolants. Specifically, it is shown that the conditioning of the reformulations using sum-of-squares interpolants does not deteriorate with increasing degrees, and problems involving sum-of-squares interpolants of hundreds of degrees can be handled without difficulty. The proposed reformulations are sufficiently well scaled that they can be solved easily with every commonly used semidefinite programming solver, such as SeDuMi, SDPT3, and CSDP. Motivating applications include convex optimization problems with semi-infinite constraints and semidefinite conic inequalities, such as those arising in the optimal design of experiments. Numerical results align with the theoretical predictions;in the problems considered, available memory was the only factor limiting the degrees of polynomials to approximately 1000.

关键词： sum-of-squares interpolation polynomial optimization semi-infinite programming design of experiments semidefinite optimization

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semi-infinite programming duality for order restricted statistical inference models

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ZOR Zeitschrift für Operations Research Methods and Models of Operations Research 1993年第3期37卷 285-301页

作者： Kortanek, K.O. Department of Management Sciences College of Business Administration The University of Iowa Iowa City 52242 Iowa 522 Phillips Hall United States

The equivalence of multinomial maximum likelihood and the isotonic projection problem: {Mathematical expression} can be established using Fenchel's Duality Theorem and subgradient and complementary slackness relationships of convex analysis, all taking place over the real numbers. In this paper non-Archimedean polynomial subgradients (Jeroslow/Kortanek '71, Blair '74, Borwein '80, and Kortanek/Soyster '81) are employed for the case where some of the observed values of the random vector are zero, corresponding to "zero counts in the traditional multinomial setting." With an appropriate linear semi-infinite programming dual pair it is shown that a vector solves the multinomial problem if and only if it converts to a solution of the isotonic projection problem. The development parallels the one of Robertson/Wright/Dykstra '88, where for the zero counts case the authors adjoin "-∞" to the real numbers and define ln(0)=-∞. © 1993 Physica-Verlag.

关键词： duality theory maximum entropy maximum likelihood non-Archimedean fields semi-infinite programming statistical inference

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GLOBAL CONVERGENCE OF A CLASS OF SMOOTH PENALTY METHODS FOR semi-infinite programming

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Journal of Systems Science & Complexity 2011年第4期24卷 769-783页

作者： Changyu WANG Haiyan ZHANG Fang LIU Institute of Operations Research Qufu Normal University Qufu 273165 China. The First Middle School of Juancheng Heze 274000 China.

For the semi-infinite programming （SIP） problem, the authors first convert it into an equivalent nonlinear programming problem with only one inequality constraint by using an integral function, and then propose a smooth penalty method based on a class of smooth functions. The main feature of this method is that the global solution of the penalty function is not necessarily solved at each iteration, and under mild assumptions, the method is always feasible and efficient when the evaluation of the integral function is not very expensive. The global convergence property is obtained in the absence of any constraint qualifications, that is, any accumulation point of the sequence generated by the algorithm is the solution of the SIP. Moreover, the authors show a perturbation theorem of the method and obtain several interesting results. Furthermore, the authors show that all iterative points remain feasible after a finite number of iterations under the Mangasarian-Fromovitz constraint qualification. Finally, numerical results are given.

关键词： Global convergence semi-infinite programming smooth penalty function perturbation function.

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Necessary optimality conditions for nonsmooth semi-infinite programming problems

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JOURNAL OF GLOBAL OPTIMIZATION 2011年第4期49卷 713-725页

作者： Kanzi, Nader PNU Dept Math Rezvanshahr E Sadugh Yazd Iran Inst Res Fundamental Sci IPM Sch Math Tehran Iran

In this paper, for a nonsmooth semi-infinite programming problem where the objective and constraint functions are locally Lipschitz, analogues of the Guignard, Kuhn-Tucker, and Cottle constraint qualifications are given. Pshenichnyi-Levin-Valadire property is introduced, and Karush-Kuhn-Tucker type necessary optimality conditions are derived.

关键词： Optimality condition semi-infinite programming Nonsmooth analysis Constraint qualification

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