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, mo...
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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.
Many classes of mixed integer nonlinear programs (MINLPs) are challenging to solve. A common approach to solve a MINLP is to use a combination of a branch-and-bound algorithm together with convexification and/or cutti...
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Many classes of mixed integer nonlinear programs (MINLPs) are challenging to solve. A common approach to solve a MINLP is to use a combination of a branch-and-bound algorithm together with convexification and/or cutting-planes. In this thesis, we develop new convexification/cutting-plane techniques for two classes of MINLPs: mixed integer conic programs and (non-convex) quadratically constrained quadratic programs. We begin by generalizing a number of important well known results in mixed inte- ger linear programming to the context of mixed integer conic programming. In particu- lar, we introduce a new class of cut generating functions and show that, under some mi- nor technical conditions, these functions, together with integer linear programming-based functions, are sufficient to yield the integer hull of intersections of conic sections in the two-dimensional space. We then focus on the representability of the convex hull of various sets derived from studying substructures of quadratically constrained quadratic programs. One of our main results shows that the convex hull of a single quadratic constraint intersected with a bounded polyhedron is second-order cone representable. For the bipartite bilinear program, a special case of the quadratically constrained quadratic program, we even design and implement an algorithm to obtain this convex hull. In addition, we introduce a new application of the bipartite bilinear program from civil engineering and report very successful computational results for this instance class. Finally, we look at the quadratically constrained quadratic program from a rank-1 per- spective, i. e., casting the non-convexity of the problem using rank-1 constraints. This ap- proach leads us to identify important substructures from which we derive and convexify several classes of sets. We then apply our results to the well-known pooling problem to obtain successful computational results.
The formulations of the generalized pooling problem (GPP) use either quality variables (the P-formulation and its variants) or split-fraction variables (the SF-formulation and its variants) to model the material balan...
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The formulations of the generalized pooling problem (GPP) use either quality variables (the P-formulation and its variants) or split-fraction variables (the SF-formulation and its variants) to model the material balance at the pools. Although the strength of the formulations among the category of the P-formulation is well understood and so is the strength of the formulations among the category of the SF-formulation, the strength of a P-formulation in comparison to a SF-formulation has not been well understood. This paper provides a formal analysis to compare the strength of the P-formulation and the SF-formulation, which leads to the conditions under which the P-formulation is at least as strong as the SF-formulation. These conditions also explain why the P-formulation is often no weaker than the SF-formulation at the root node of branch-and-bound search. The theoretical results are verified by a computational study of 26 problem instances. (C) 2022 Elsevier Ltd. All rights reserved.
We propose a numerical method to compute stabilizing state feedback control laws and associated polyhedral invariant sets for nonlinear systems represented by Fuzzy Takagi-Sugeno (T-S) models, subject to state and con...
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We propose a numerical method to compute stabilizing state feedback control laws and associated polyhedral invariant sets for nonlinear systems represented by Fuzzy Takagi-Sugeno (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.
This paper studies the widely applied fixed charge network ow problem (FCNFP), which is NP-hard. We approximate the FCNFP with a bilinear programming problem that is determined by a parameter ɛ. When ɛ is small enough...
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This paper studies the widely applied fixed charge network ow problem (FCNFP), which is NP-hard. We approximate the FCNFP with a bilinear programming problem that is determined by a parameter ɛ. When ɛ 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 ɛ. In this paper, these bilinear programming problems are solved by alternately solving two coupled linear programming problems. A dynamic method is proposed to update ɛ 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.
Back-propagation in neural networks has difficulties that(i) there are several sensitive *** affect the speed and precision of learning,such as the slope of sigmoidal function,learning coefficients and initial values ...
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Back-propagation in neural networks has difficulties that(i) there are several sensitive *** affect the speed and precision of learning,such as the slope of sigmoidal function,learning coefficients and initial values of decision *** so on.(ii) it tends to be trapped in local *** this *** methods which overcome these difficulties will be reported: In order to overcome the point(ii) stated above,several stochastic methods such as random search and simulated annealing have been *** this paper,however,we propose a method using GA(Genetic Algorithm) for getting rid of the above two difficulties simultaneously. This approach provides a good estimate of parameters and intial values for getting global minima in back-propagation. Next,as a deterministic approach,a bilinear programming model for back-propagation is *** this formulation,parameters like the slope of sigmoidal function are automaticlay optimizied and moreover the obtained solution can be guaranteed to be a global minima of the error function.
Two structured classes of zero-sum two-person finite (state and action spaces) semi-Markov games with discounted payoffs, namely, Additive Reward-Additive Transition and Action Independent Transition Time (AR-AT-AITT)...
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Two structured classes of zero-sum two-person finite (state and action spaces) semi-Markov games with discounted payoffs, namely, Additive Reward-Additive Transition and Action Independent Transition Time (AR-AT-AITT) and Additive Reward-Action independent Transition and Additive Transition Time (AR-MT-ATT) have been studied. We propose two practical situations of economic competition viz. petroleum game and groundwater game that suitably fit into such classes of games, respectively. Solution (value and pure stationary optimals) to such classes of games can be derived from optimal solution to appropriate bilinear programs with linear constraints. We present a stepwise generalized principal pivoting algorithm for solving the vertical linear cornplementarity problem (VLCP) obtained from such game problems. Moreover, a neural network dynamics is proposed for solving such structured classes. Examples are worked out. to compare the usefulness of the above three algorithms.
In this paper we present an algorithm for the pooling problem in refinery optimization based on a bilinear programming approach. The pooling problem occurs frequently in process optimization problems, especially refin...
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作者：
Liberti, LeoKucherenko, SergeiDEI
Politecnico di Milano Milano 20133 P.zza L. da Vinci 32 Italy CPSE
Imperial College London London SW7 2BY United Kingdom
In this paper, we compare two different approaches to nonconvex global optimization. The first one is a deterministic spatial Branch-and-Bound algorithm, whereas the second approach is a Quasi Monte Carlo (QMC) varian...
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