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THE ADAPTIVE CONVEXIFICATION ALGORITHM: A FEASIBLE POINT METHOD FOR SEMI-INFINITE programMING

SIAM JOURNAL ON OPTIMIZATION 2008年第4期18卷 1187-1208页

作者： Floudas, Christodoulos A. Stein, Oliver Univ Karlsruhe Sch Econ & Business Engn D-76128 Karlsruhe Germany Princeton Univ Dept Chem Engn Princeton NJ 08540 USA

We present a new numerical solution method for semi-infinite optimization problems. Its main idea is to adaptively construct convex relaxations of the lower level problem, replace the relaxed lower level problems equivalently by their Karush-Kuhn-Tucker conditions, and solve the resulting mathematical programs with complementarity constraints. This approximation produces feasible iterates for the original problem. The convex relaxations are constructed with ideas from the alpha BB method of global optimization. The necessary upper bounds for second derivatives of functions on box domains can be determined using the techniques of interval arithmetic, where our algorithm already works if only one such bound is available for the problem. We show convergence of stationary points of the approximating problems to a stationary point of the original semi-infinite problem within arbitrarily given tolerances. Numerical examples from Chebyshev approximation and design centering illustrate the performance of the method.

关键词： semi-infinite programming alpha BB global optimization convex optimization mathematical program with complementarity constraints bilevel optimization

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Directional lane-allocation concerning demand uncertainty

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IET INTELLIGENT TRANSPORT SYSTEMS 2013年第4期7卷 380-387页

作者： Zhang, Lihui Zhong, Shaopeng Dalian Univ Technol Sch Transportat & Logist Dalian 116024 Liaoning Provin Peoples R China

Regarding the one-way restriction as a special case of the lane-allocation operation, this study formulates the general lane-allocation problem as a mathematical program with complementarity constraints (MPCC). Recognising the day-to-day and within-day travel demand variations, another counterpart of the MPCC is formulated to obtain a robust lane-allocation scheme that performs better against a set of worst-case demand scenarios. Solution algorithms based on active sets are proposed to address the problems by sequentially solving some relaxed-MPCC and a binary knapsack problem. Numerical examples are presented to validate the proposed model and algorithm.

关键词： knapsack problems mathematical programming road traffic directional lane-allocation demand uncertainty lane-allocation operation mathematical program with complementarity constraints MPCC within-day travel demand variations robust lane-allocation scheme worst-case demand scenarios active sets binary knapsack problem

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DEREGULATED ELECTRICITY MARKETS WITH THERMAL LOSSES AND PRODUCTION BOUNDS: MODELS AND OPTIMALITY CONDITIONS

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RAIRO-OPERATIONS RESEARCH 2016年第1期50卷 19-38页

作者： Aussel, Didier Cervinka, Michal Marechal, Matthieu Univ Perpignan Lab PROMES UPR CNRS 8521 Rambla Thermodynam F-66100 Perpignan France Acad Sci Czech Republ Inst Informat Theory & Automat Pod Vodarenskou Vezi 4 CR-18208 Prague Czech Republic Charles Univ Prague Fac Social Sci Inst Econ Studies Opletalova 26 Prague 11000 Czech Republic Ctr Modelamiento Matemat Santiago Chile

A multi-leader-common-follower game formulation has been recently used by many authors to model deregulated electricity markets. In our work, we first propose a model for the case of electricity market with thermal losses on transmission and with production bounds, a situation for which we emphasize several formulations based on different types of revenue functions of producers. Focusing on a problem of one particular producer, we provide and justify an MPCC reformulation of the producer's problem. Applying the generalized differential calculus, the so-called M-stationarity conditions are derived for the reformulated electricity market model. Finally, verification of suitable constraint qualification that can be used to obtain first order necessary optimality conditions for the respective MPCCs are discussed.

关键词： Deregulated electricity market production bounds mathematical program with complementarity constraints M-stationarity calmness

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OPTIMAL CONTROL PROBLEMS WITH TERMINAL complementarity constraints

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SIAM JOURNAL ON OPTIMIZATION 2018年第4期28卷 3079-3104页

作者： Benita, Francisco Mehlitz, Patrick Singapore Univ Technol & Design Dept Engn Syst & Design Singapore 487372 Singapore Brandenburg Tech Univ Cottbus Senftenberg Chair Optimal Control D-03046 Cottbus Germany

In this paper, we study an optimal control problem of ordinary differential equations with linear dynamics, affine mixed control-state constraints, and terminal complementarity constraints on the state function. We derive its weak, Mordukhovich, and strong stationarity conditions, and we present constraint qualifications which ensure that these conditions are satisfied at a locally optimal solution of the optimal control problem.

关键词： mathematical program with complementarity constraints optimal control optimality conditions programming in Banach spaces W-, M-, S-stationarity

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On the convergence properties of modified augmented Lagrangian methods for mathematical programming with complementarity constraints

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JOURNAL OF GLOBAL OPTIMIZATION 2010年第2期46卷 217-232页

作者： Luo, H. Z. Sun, X. L. Xu, Y. F. Wu, H. X. Fudan Univ Sch Management Dept Management Sci Shanghai 200433 Peoples R China Zhejiang Univ Technol Dept Appl Math Hangzhou 310032 Zhejiang Peoples R China Hangzhou Dianzi Univ Dept Math Hangzhou 310018 Zhejiang Peoples R China

In this paper, we present new convergence results of augmented Lagrangian methods for mathematical programs with complementarity constraints (MPCC). Modified augmented Lagrangian methods based on four different algorithmic strategies are considered for the constrained nonconvex optimization reformulation of MPCC. We show that the convergence to a global optimal solution of the problem can be ensured without requiring the boundedness condition of the multipliers.

关键词： mathematical program with complementarity constraints Modified augmented Lagrangian methods Nonconvex constrained optimization Convergence to global solution

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Optimality conditions and newton-type methods for mathematical programs with vanishing constraints

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COMPUTATIONAL MATHEMATICS AND mathematical PHYSICS 2009年第7期49卷 1128-1140页

作者： Izmailov, A. F. Pogosyan, A. L. Moscow MV Lomonosov State Univ Fac Computat Math & Cybernet Moscow 119992 Russia

A new class of optimization problems is discussed in which some constraints must hold in certain regions of the corresponding space rather than everywhere. In particular, the optimal design of topologies for mechanical structures can be reduced to problems of this kind. Problems in this class are difficult to analyze and solve numerically because their constraints are usually irregular. Some known first- and second-order necessary conditions for local optimality are refined for problems with vanishing constraints, and special Newton-type methods are developed for solving such problems.

关键词： mathematical program with vanishing constraints mathematical program with complementarity constraints constraint qualification optimality conditions sequential quadratic programming active-set method

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mathematical programs with complementarity constraints in Banach Spaces

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2015年第2期166卷 480-507页

作者： Wachsmuth, Gerd Tech Univ Chemnitz Fac Math Professorship Numer Math Partial Differential Equ D-09107 Chemnitz Germany

We consider optimization problems in Banach spaces involving a complementarity constraint, defined by a convex cone K. By transferring the local decomposition approach, we define strong stationarity conditions and provide a constraint qualification, under which these conditions are necessary for optimality. To apply this technique, we provide a new uniqueness result for Lagrange multipliers in Banach spaces. In the case that the cone K is polyhedral, we show that our strong stationarity conditions possess a reasonable strength. Finally, we generalize to the case where K is not a cone and apply the theory to two examples.

关键词： Strong stationarity mathematical program with complementarity constraints Polyhedricity Optimality conditions

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A semismooth sequential quadratic programming method for lifted mathematical programs with vanishing constraints

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COMPUTATIONAL MATHEMATICS AND mathematical PHYSICS 2011年第6期51卷 919-941页

作者： Izmailov, A. F. Pogosyan, A. L. Moscow MV Lomonosov State Univ Fac Computat Math & Cybernet Moscow 119991 Russia

mathematical programs with vanishing constraints are a difficult class of optimization problems with important applications to optimal topology design problems of mechanical structures. Recently, they have attracted increasingly more attention of experts. The basic difficulty in the analysis and numerical solution of such problems is that their constraints are usually nonregular at the solution. In this paper, a new approach to the numerical solution of these problems is proposed. It is based on their reduction to the so-called lifted mathematical programs with conventional equality and inequality constraints. Special versions of the sequential quadratic programming method are proposed for solving lifted problems. Preliminary numerical results indicate the competitiveness of this approach.

关键词： mathematical program with vanishing constraints lifted problem mathematical program with complementarity constraints constraint qualifications optimality conditions sequential quadratic programming

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Global resolution of the support vector machine regression parameters selection problem with LPCC

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EURO JOURNAL ON COMPUTATIONAL OPTIMIZATION 2015年第3期3卷 197-261页

作者： Lee, Yu-Ching Pang, Jong-Shi Mitchell, John E. Univ Illinois Urbana IL 61801 USA Univ Southern Calif Los Angeles CA 90089 USA Rensselaer Polytech Inst Troy NY 12180 USA

Support vector machine regression is a robust data fitting method to minimize the sum of deducted residuals of regression, and thus is less sensitive to changes of data near the regression hyperplane. Two design parameters, the insensitive tube size (ee) and the weight assigned to the regression error trading off the normed support vector (Ce), are selected by user to gain better forecasts. The global training and validation parameter selection procedure for the support vector machine regression can be formulated as a bi-level optimization model, which is equivalently reformulated as linear program with linear complementarity constraints (LPCC). We propose a rectangle search global optimization algorithm to solve this LPCC. The algorithm exhausts the invariancy regions on the parameter plane ((Ce, ee)-plane) without explicitly identifying the edges of the regions. This algorithm is tested on synthetic and real-world support vector machine regression problems with up to hundreds of data points, and the efficiency are compared with several approaches. The obtained global optimal parameter is an important benchmark for every other selection of parameters.

关键词： Support vector Machine regression Parameter selection global optimal parameter mathematical program with complementarity constraints Global optimization algorithm

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Closed-loop Formulation for Nonlinear Dynamic Real-time Optimization

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IFAC-PapersOnLine 2016年第7期49卷 406-411页

作者： Jamaludin, Mohammad Zamry Swartz, Christopher L.E. Department of Chemical Engineering McMaster University 1280 Main St. West HamiltonONL8S 4L7 Canada

Dynamic real-time optimization (DRTO) is a higher level online strategy that exploits plant economic potential by making appropriate adjustments to the lower level controller set-point trajectories. In this work, we propose a closed-loop formulation for a nonlinear DRTO calculation in the form of a bilevel programming problem. A nonlinear differential algebraic equation (DAE) system that describes the process dynamic behavior is utilized with an embedded constrained predictive control (MPC) optimization subproblem to generate the approximate closed-loop response dynamics at the primary economic optimization layer. The bilevel DRTO problem is reformulated as a single-level mathematical program with complementarity constraints (MPCC) by replacing the MPC optimization subproblem by its Karush-Kuhn-Tucker (KKT) optimality conditions. We investigate the economics and control performance of the proposed strategy based on a polymer grade transition case study in the presence of plant-model mismatch and a disturbance, and a comparison is made with the application of a linear DRTO prediction model. © 2016

关键词： Constrained optimization Differential equations Economics Model predictive control Nonlinear equations Predictive control systems Back off mechanisms Bilevel programming problem complementarity constraint Economic optimization Karush Kuhn Tucker (KKT) mathematical program with complementarity constraints Nonlinear differential algebraic equations Real time optimization

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