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A GLOBAL OPTIMIZATION ALGORITHM FOR LINEAR FRACTIONAL AND bilinear PROGRAMS

JOURNAL OF GLOBAL OPTIMIZATION 1995年第1期6卷 39-76页

作者： QUESADA, I GROSSMANN, IE CARNEGIE MELLON UNIV DEPT CHEM ENGN PITTSBURGH PA 15213 USA

In this paper a deterministic method is proposed for the global optimization of mathematical programs that involve the sum of linear fractional and/or bilinear terms. Linear and nonlinear convex estimator functions are developed for the linear fractional and bilinear terms. Conditions under which these functions are nonredundant are established. It is shown that additional estimators can be obtained through projections of the feasible region that can also be incorporated in a convex nonlinear underestimator problem for predicting lower bounds for the global optimum. The proposed algorithm consists of a spatial branch and bound search for which several branching rules are discussed. Illustrative examples and computational results are presented to demonstrate the efficiency of the proposed algorithm.

关键词： NONCONVEX OPTIMIZATION bilinear programming FRACTIONAL programming CONVEX UNDERESTIMATORS

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A SIMPLE bilinear OPTIMIZATION PROBLEM

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SYSTEMS & CONTROL LETTERS 1986年第1期8卷 1-4页

作者： WITSENHAUSEN, HS AT&T BELL LABS MURRAY HILLNJ 07974

Multilinear optimization problems arise in many contexts, yet already the bilinear case, even for variables restricted to parallelotopes, is intractable in general. As an example of a manageable case, which may also be used to bound other cases, the elementary problem of maximizing the inner product of two vectors, each constrained to a sphere, is solved here.

关键词： Complexity bilinear programming Multilinear programming Maximum inner product Two-sphere problem

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Strong formulations for the pooling problem

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JOURNAL OF GLOBAL OPTIMIZATION 2013年第3期56卷 897-916页

作者： Alfaki, Mohammed Haugland, Dag Univ Bergen Dept Informat N-5020 Bergen Norway

The pooling problem is a well-studied global optimization problem with applications in oil refining and petrochemical industry. Despite the strong NP-hardness of the problem, which is proved formally in this paper, most instances from the literature have recently been solved efficiently by use of strong formulations. The main contribution from this paper is a new formulation that proves to be stronger than other formulations based on proportion variables. Moreover, we propose a promising branching strategy for the new formulation and provide computational experiments confirming the strength of the new formulation and the effectiveness of the branching strategy.

关键词： Pooling problem bilinear programming Global optimization Linear relaxation Computational complexity

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SUM OF SQUARES CERTIFICATES FOR CONTAINMENT OF H-POLYTOPES IN ν-POLYTOPES

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SIAM JOURNAL ON DISCRETE MATHEMATICS 2016年第2期30卷 763-776页

作者： Kellner, Kai Theobald, Thorsten Goethe Univ Frankfurt FB Inst Math 12 Postfach 11 19 32 D-60054 Frankfurt Germany

Given an H-polytope P and a nu-polytope Q, the decision problem of whether P is contained in Q is co-NP-complete. This hardness remains if P is restricted to be a standard cube and Q is restricted to be the affine image of a cross polytope. While this hardness classification by Freund and Orlin dates back to 1985, for general dimension there seems to be only limited progress on the decision problem so far. Based on a formulation of the problem in terms of a bilinear feasibility problem, we study sum of squares certificates to decide the containment problem. These certificates can be computed by a semidefinite hierarchy. As a main result, we show that under mild and explicitly known preconditions the semidefinite hierarchy converges in finitely many steps. In particular, if P is contained in a large nu-polytope Q (in a well-defined sense), then containment is certified by the first step of the hierarchy.

关键词： polytope containment sum of squares semidefinite hierarchy bilinear programming Positivstellensatz

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Robust Positively Invariant Polyhedral Sets and Constrained Control using Fuzzy T-S Models: a bilinear Optimization Design Strategy 21st

Robust Positively Invariant Polyhedral Sets and Constrained ...

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21st IFAC World Congress on Automatic Control - Meeting Societal Challenges

作者： Dorea, Carlos E. T. Castelan, Eugenio B. Ernesto, Jackson G. Univ Fed Rio Grande do Norte UFRN CT DCA BR-59078280 Natal RN Brazil Univ Fed Santa Catarina UFSC DAS Florianopolis SC Brazil

We propose a numerical method to compute stabilizing state feedback control laws and associated polyhedral invariant sets for nonlinear systems represented by Fuzzy TakagiSugeno (T-S) models, subject to state and control constraints, and persistent disturbances. Sufficient conditions are derived under which a given polyhedral set is positively invariant under a Parallel Distributed Compensation (PDC), in the form of bilinear algebraic inequalities. Then, a bilinear programming (BP) problem is proposed to compute the state feedback gains and an associated positively invariant polyhedron, with predefined complexity, which solve a constrained regulation problem for the Fuzzy T-S system. A numerical example illustrates the effectiveness of the method. Copyright (C) 2020 The Authors.

关键词： Fuzzy T-S models Parallel Distributed Compensation Constrained Control Invariant Sets bilinear programming

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A MATHEMATICAL-programming MODEL TO CONSIDER ALTERNATIVE CROPS

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OMEGA-INTERNATIONAL JOURNAL OF MANAGEMENT SCIENCE 1991年第2-3期19卷 169-179页

作者： ETO, H UNIV TSUKUBA INST SOCIOECON PLANNINGSAKURAIBARAKI 305JAPAN

Japan's increased import of grain in the last decades has brought about the need for Japanese farmers to convert from rice production to other crops in which they are unexperienced. This paper presents a model and analyses to find effective means for inducing the farmers to convert in an acceptable way. The model is composed of several subproblems to represent different interest groups. The farmer's subproblems are to maximise their revenues whereas the government subproblem is to save expenditure in attaining the conversion. These subproblems are coordinated from a holistic point of view by a coordination subproblem of goal programming on equity among farms and on the soundness of government budget. Part of computation was carried out interactively with the participation of experts.

关键词： AGRICULTURAL POLICY SUBSIDY INTERACTIVE METHOD GOAL programming DISJUNCTIVE programming bilinear programming

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Interval Evaluation of Stationary State Probabilities for Markov Set-Chain Models 10

Interval Evaluation of Stationary State Probabilities for Ma...

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10th International Conference on Advanced Computer Information Technologies (ACIT)

作者： Lyubchyk, Leonid Grinberg, Galyna Lubchick, Maria Galuza, Alexey Akhiiezer, Olena Natl Tech Univ Dept Comp Math & Data Anal Kharkiv Polytech Inst Kharkiv Ukraine Natl Tech Univ Kharkiv Polytech Inst Dept Econ Cybernet & Management Kharkiv Ukraine

ISBN: (纸本)9781728167602

Interval estimation problem for stationary state probability distribution of set-chain Markov model uncertainties of transition matrix parameters is considered. To obtain final probability vector interval estimates, an optimization approach is proposed using regularized Lagrange function. To solve the obtained regularized bilinear programming problem, computational gradient algorithms are used with ensures the stability of resulting estimates. An example of Markov model of a Bonus-Malus system with interval uncertainties of claim flow intensity is presented.

关键词： bilinear programming Bonus-Malus interval estimate Lagrange function Markov chain regularization set-chain transition matrix

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Explicit Computation of Stabilizing Feedback Control Gains Using Polyhedral Lyapunov Functions 23

Explicit Computation of Stabilizing Feedback Control Gains U...

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IEEE International Conference on Automation (ICA) / 23rd Congress of the Chilean-Association-of-Automatic-Control (ACCA)

作者： Briao, Stephanie L. Pedrosa, Matheus V. A. Castelan, Eugenio B. Camponogara, Eduardo de Assis, Leonardo S. Univ Fed Santa Catarina Dept Automat & Syst Engn Florianopolis SC Brazil

ISBN: (纸本)9781538655863

The explicit computation of stabilizing Static Output Feedback (SOF) and State Feedback (StF) control gains is treated in this work by considering polyhedral Lyapunov functions (PLF). More specifically, the associated nonlinear algebraic necessary and sufficient conditions that prove closed-loop stability are exploited to propose bilinear optimization problems that allow to explicitly design the control gains. The objective function considered here in allows to weigh the speed of states convergence and the control effort or the size of a region of admissible initial conditions in case of constrained control inputs. Furthermore, the proposed design for optimization problems can be implemented by using an existing non-linear optimization solver that shows to be specially efficient to deal with the considered bilinearities. A numerical example is presented to show the effectiveness of our proposal for both SOF and StF designs. Some comments about future research directions using the considered framework are also provided.

关键词： bilinear programming Discrete-time systems Explicit gains computation Polyhedral Lyapunov functions State feedback Static output feedback

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A Dynamic Method to Solve the Fixed Charge Network Flow Problem 21st

A Dynamic Method to Solve the Fixed Charge Network Flow Prob...

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21st IFAC World Congress on Automatic Control - Meeting Societal Challenges

作者： Nie, Zhibin Wang, Shuning Tsinghua Univ Dept Automat Natl Lab Informat Sci & Technol Beijing 100084 Peoples R China

This paper studies the widely applied fixed charge network flow problem (FCNFP), which is NP-hard. We approximate the FCNFP with a bilinear programming problem that is determined by a parameter epsilon. When epsilon is small enough, the optimal solution to the bilinear programming problem is the same as the optimal solution to the FCNFP. Therefore, solving the FCNFP can be transformed into solving a series of bilinear programming problems with decreasing epsilon. In this paper, these bilinear programming problems are solved by alternately solving two coupled linear programming problems. A dynamic method is proposed to update epsilon after solving one of the linear programming problems rather than solving the whole bilinear programming problem. Numerical experiments show the performance of the proposed method. Copyright (C) 2020 The Authors.

关键词： Network flow problem Fixed charge bilinear programming Dynamic method

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An Improved 3D bilinear Multidimensional Morphable Models Used in 3D Face Recognition

An Improved 3D Bilinear Multidimensional Morphable Models Us...

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International Conference on Information Science, Electronics and Electrical Engineering (ISEEE)

作者： Wang, Liying Liu, Bixia Su, Songzhi Cheng, Yun Li, Shaozi Xiamen Univ Sch Informat Sci & Engn Xiamen 361005 Peoples R China Hunan Univ Humanities Sci & Technol Loudi Dept Commun & Control engn Loudi Hunan Peoples R China

ISBN: (纸本)9781479931972

This paper proposes the improved 3D bilinear Multidimensional Morphable Models (BMMMs) which can be used in face recognition, etc. In each level of the multidimensional model, we introduce the bilinear programming model for the model matching problem as well as a globally optimal algorithm for bilinear programs. Both tight convex relaxations of the objective function and a convergent branching strategy contribute to the success of the globally optimal algorithm for bilinear programs. Experiments with synthetic data validate that the 3D bilinear Multidimensional Morphable Models outperform the 3D Morphable Models in optimality, model matching speed, convergence rate and robustness to noise and outliers.

关键词： bilinear programming 3D Multidimensional Morphable Models Branch and Bound

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