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Is Pessimistic Bilevel programming a Special Case of a mathematical Program with complementarity constraints?

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第2期181卷 504-520页

作者： Aussel, Didier Svensson, Anton Univ Perpignan Perpignan France Univ Chile Santiago Chile

One of the most commonly used methods for solving bilevel programming problems (whose lower level problem is convex) starts with reformulating it as a mathematical program with complementarity constraints. This is done by replacing the lower level problem by its Karush-Kuhn-Tucker optimality conditions. The obtained mathematical program with complementarity constraints is (locally) solved, but the question of whether a solution of the reformulation yields a solution of the initial bilevel problem naturally arises. The question was first formulated and answered negatively, in a recent work of Dempe and Dutta, for the so-called optimistic approach. We study this question for the pessimistic approach also in the case of a convex lower level problem with a similar answer. Some new notions of local solutions are defined for these minimax-type problems, for which the relations are shown. Some simple counterexamples are given.

关键词： Bilevel problem mathematical programming with complementarity constraints Pessimistic approach

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complementarity model for load tap changing transformers in stability based OPF problem

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ELECTRIC POWER SYSTEMS RESEARCH 2006年第6-7期76卷 592-599页

作者： Roman, C Rosehart, W Univ Calgary Dept Elect & Comp Engn Calgary AB T2N 1N4 Canada

The paper presents a stability optimal power flow (OPF) based problem that incorporates complementarity models for load tap changing transformers (LTCs). The complementarity formulation models the change of behavior of LTC controlled buses when the tap becomes blocked at one of its limits and the voltage level can no longer be regulated. The incorporation of proposed model in stability OPF based problem leads to a more accurate estimation of the maximum or critical loading level. (c) 2005 Elsevier B.V. All rights reserved.

关键词： voltage stability load tap changing transformers optimal power flow mathematical programming with complementarity constraints

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Feasibility problems with complementarity constraints

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2016年第1期249卷 41-54页

作者： Andreani, R. Judice, J. J. Martinez, J. M. Martini, T. Univ Estadual Campinas Dept Appl Math IMECC UNICAMP BR-13083859 Campinas SP Brazil Univ Coimbra Inst Telecomun Coimbra Portugal

A Projected-Gradient Underdetermined Newton-like algorithm will be introduced for finding a solution of a Horizontal Nonlinear complementarity Problem (HNCP) corresponding to a feasible solution of a mathematical programming Problem with complementarity constraints (MPCC). The algorithm employs a combination of Interior-Point Newton-like and Projected-Gradient directions with a line-search procedure that guarantees global convergence to a solution of HNCP or, at least, a stationary point of the natural merit function associated to this problem. Fast local convergence will be established under reasonable assumptions. The new algorithm can be applied to the computation of a feasible solution of MPCC with a target objective function value. Computational experience on test problems from well-known sources will illustrate the efficiency of the algorithm to find feasible solutions of MPCC in practice. (C) 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved.

关键词： Global optimization Nonlinear programming Nonlinear systems of equations complementarity problems mathematical programming with complementarity constraints

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A multi-objective reactive distillation optimization model for Fischer-Tropsch synthesis

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COMPUTERS & CHEMICAL ENGINEERING 2020年第0期135卷 106754-000页

作者： Zhang, Yizu He, Naien Masuku, Cornelius M. Biegler, Lorenz T. Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA Univ South Africa Dept Civil & Chem Engn Private Bag X6 ZA-1710 Florida South Africa

In the design of a reactive distillation column, aspects such as column configuration, catalyst loading, tray temperature, and side extraction rates should be well considered. Though preferences in Fischer-Tropsch (FT) synthesis may vary, it is acknowledged that the final product contains a wide range of hydrocarbons including fuel gas, gasoline, diesel, and linear wax. Zhang et al. (2018) previously developed an equation-oriented framework for optimal synthesis of integrated reactive distillation systems for FT processes. Here, we extend the mass, equilibrium, summation, and heat equations to a mathematical program with complementarity constraints to deal with possible dry trays in the non-reactive sections. The purpose of describing disappearing phases is to avoid infeasibilities due to multiple bilinear terms in the model for complicated model structures. The model is implemented by solving initialization steps and a sequence of nonlinear programming problems to determine an optimal structure and operating conditions. Design specifications for multiple products could be set as individual objectives to determine design limits. Moreover, a balance of multiple objectives could be reached by formulating the reactive distillation model as a multi-objective optimization problem. In this work, we employ the augmented epsilon-constraint method. The results show that significant design insights can be gained from the Pareto-optimal front regarding acceptable trade-offs among various objectives. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Equation-oriented optimization framework mathematical programming with complementarity constraints Multi-objective optimization Low temperature Fischer-Tropsch Process intensification

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Nonlinear robust optimization via sequential convex bilevel programming

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mathematical programming 2013年第1-2期142卷 539-577页

作者： Houska, Boris Diehl, Moritz Katholieke Univ Leuven Optimizat Engn Ctr OPTEC B-3001 Louvain Belgium

In this paper, we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min-max problems which arise in the context of semi-infinite programming. Here, our main motivation are nonlinear inequality constrained robust optimization problems. In the first part of the paper, we propose a conservative approximation strategy for such nonlinear and non-convex robust optimization problems: under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given, we show how to formulate a lower-level concave min-max problem which approximates the robust counterpart in a conservative way. This approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties. In the second part of the paper, we review existing theory on optimality conditions for nonlinear lower-level concave min-max problems which arise in the context of semi-infinite programming. Regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem, we end up with a structured mathematical program with complementarity constraints (MPCC). The special hierarchical structure of this MPCC can be exploited in a novel sequential convex bilevel programming algorithm. We discuss the surprisingly strong global and locally quadratic convergence properties of this method, which can in this form neither be obtained with existing SQP methods nor with interior point relaxation techniques for general MPCCs. Finally, we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example.

关键词： Robust optimization mathematical programming with complementarity constraints Bilevel optimization Semi-infinite optimization Sequential convex programming

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The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets

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mathematical programming 2012年第1期136卷 183-207页

作者： Stein, Oliver Steuermann, Paul KIT Karlsruhe Germany

A numerical solution method for semi-infinite optimization problems with arbitrary, not necessarily box-shaped, index sets is presented. Following the ideas of Floudas and Stein (SIAM J Optim 18:1187-1208, 2007), convex relaxations of the lower level problem are adaptively constructed and then reformulated as mathematical programs with complementarity constraints and solved. Although the index set is arbitrary, this approximation produces feasible iterates for the original problem. The convex relaxations and needed parameters are constructed with ideas of the alpha BB method of global optimization and interval methods. It is shown that after finitely many steps an -stationary point of the original semi-infinite problem is reached. A numerical example illustrates the performance of the proposed method.

关键词： Semi-infinite programming alpha BB Global optimization Convex optimization mathematical programming with complementarity constraints Bilevel optimization

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Analysis of control models based on dFBA for fed-batch bioreactors solved by interior-point methods 13th

Analysis of control models based on dFBA for fed-batch biore...

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13th IFAC Symposium on Dynamics and Control of Process Systems, including Biosystems (DYCOPS)

作者： Nakama, Caroline S. M. Jaschke, Johannes Norwegian Univ Sci & Technol NTNU Dept Chem Engn Trondheim Norway

Model-based control of bioprocesses based on dynamic flux balance analysis (dFBA) is an interesting strategy due to the possibility of accounting for wider ranges of cellular behaviour and operation conditions than the one based on unstructured models. This type of control models are bi-level optimization problems, since dFBA comprises of differential equations and a linear programming (LP) model. They can be solved with a nonlinear programming (NLP) solver by replacing the LP model with its first-order optimality conditions (KKT conditions) and discretizing the differential equations. When following this approach, it is important to carefully design the optimization problem considering properties of the selected solver. In this work, we show that we can formulate and solve model-based control models with dFBA for fed-batch bioprocesses by using the KKT conditions of the LP model, relaxing the inequality-multiplier complementarities that arise from these conditions and employing a line search interior point solver. We demonstrate this process on a case study that seeks to the maximize growth of Escherichia coli on glucose and discuss fundamental steps that one should carefully evaluate, such as constraint dependence, handling of complementarities and model initialization. Copyright (C) 2022 The Authors.

关键词： Bioprocess control dynamic flux balance analysis mathematical programming with complementarity constraints bi-level optimization dynamic optimization

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Global Product Design Platforming: A Comparison of Two Equilibrium Solution Methods

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JOURNAL OF MECHANICAL DESIGN 2023年第6期145卷 061702-061702页

作者： Case, Sarah Michalek, Jeremy J. Whitefoot, Kate S. Carnegie Mellon Univ Mech Engn Pittsburgh PA 15213 USA Carnegie Mellon Univ Mech Engn Engn & Publ Policy Civil & Environm Engn by Courtesy Pittsburgh PA 15213 USA Carnegie Mellon Univ Mech Engn Engn & Publ Policy Pittsburgh PA 15213 USA

Global product platforms can reduce production costs through economies of scale and learning but may decrease revenues by restricting the ability to customize for each market. We model the global platforming problem as a Nash equilibrium among oligopolistic competing firms, each maximizing its profit across markets with respect to its pricing, design, and platforming decisions. We develop and compare two methods to identify Nash equilibria: (1) a sequential iterative optimization (SIO) algorithm, in which each firm solves a mixed-integer nonlinear programming problem globally, with firms iterating until convergence;and (2) a mathematical program with equilibrium constraints (MPEC) that solves the Karush Kuhn Tucker conditions for all firms simultaneously. The algorithms' performance and results are compared in a case study of plug-in hybrid electric vehicles where firms choose optimal battery capacity and whether to platform or differentiate battery capacity across the US and Chinese markets. We examine a variety of scenarios for (1) learning rate and (2) consumer willingness to pay (WTP) for range in each market. For the case of two firms, both approaches find the Nash equilibrium in all scenarios. On average, the SIO approach solves 200 times faster than the MPEC approach, and the MPEC approach is more sensitive to the starting point. Results show that the optimum for each firm is to platform when learning rates are high or the difference between consumer willingness to pay for range in each market is relatively small. Otherwise, the PHEVs are differentiated with low-range for China and high-range for the US.

关键词： product platform learning by doing design for market systems designoptimization mixed-integer nonlinear programing global optimization mathematical programming with equilibrium constraints mathematical programming with complementarity constraints Nash equilibrium oligopoly product design product familydesign product platform design

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Analysis of control models based on dFBA for fed-batch bioreactors solved by interior-point methods

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IFAC-PapersOnLine 2022年第7期55卷 131-136页

作者： Caroline S.M. Nakama Johannes Jäschke

关键词： Bioprocess control dynamic flux balance analysis mathematical programming with complementarity constraints bi-level optimization dynamic optimization

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