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Metaheuristics for variable-size mixed optimization problems: A unified taxonomy and survey

SWARM AND EVOLUTIONARY COMPUTATION 2024年 89卷

作者： Talbi, El-Ghazali Univ Lille CNRS Inria Cent LilleUMR 9189 CRIStAL F-59000 Lille France

Many real world optimization problems are formulated as mixed-variable optimization problems (MVOPs) which involve both continuous and discrete variables. MVOPs including dimensional variables are characterized by a variable-size search space. Depending on the values of dimensional variables, the number and type of the variables of the problem can vary dynamically. MVOPs and variable-size MVOPs (VMVOPs) are difficult to solve and raise a number of scientific challenges in the design of metaheuristics. Standard metaheuristics have been first designed to address continuous or discrete optimization problems, and are not able to tackle VMVOPs in an efficient way. The development of metaheuristics for solving such problems has attracted the attention of many researchers and is increasingly popular. However, to our knowledge there is no well established taxonomy or comprehensive survey for handling this important family of optimization problems. This paper presents an unified taxonomy for metaheuristic solutions for solving VMVOPs in an attempt to provide a common terminology and classification mechanisms. It provides a general mathematical formulation and concepts of VMVOPs, and identifies the various solving methodologies than can be applied in metaheuristics. The advantages, the weaknesses and the limitations of the presented methodologies are discussed. The proposed taxonomy also allows to identify some open research issues which needs further in-depth investigations.

关键词： Metaheuristics mixed-variable optimization problem variable-size mixed-variable optimization problem mixed optimization mixed variable programming variable-space design Decomposition-based optimization

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Unit commitment in oligopolistic markets by nonlinear mixed variable programming

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OPTIMIZATION AND ENGINEERING 2010年第2期11卷 213-245页

作者： Casolino, Giovanni Mercurio Liuzzi, Giampaolo Losi, Arturo CNR Ist Anal Sistemi & Informat A Ruberti Vle Manzoni 30 I-00185 Rome Italy Univ Cassino Dipartimento Ingn Ind I-03043 Cassino Fr Italy

In the paper we consider the unit commitment problem in oligopolistic markets. The formulation of the problem involves both integer and continuous variables and nonlinear functions as well, thus yielding a nonlinear mixed variable programming problem. Our formulation takes into account all technical constraints for the generating units, such as ramp rate and minimum up and down time constraints, considers the uncertainty related to the selling prices and allows modeling their dependence on the total output of a producer. The objective function is the expected value of the revenue over the different scenarios minus a term which takes into account the risk related to the decision. To solve the problem we adopt a recently proposed method for mixed integer nonlinear programming problems and use a derivative free algorithm to solve the continuous subproblems. We report results for two operators: one managing a single unit and the other managing three units. Numerical results give evidence to the features of the modeling and show viability of the adopted algorithm.

关键词： Unit commitment Oligopolistic markets mixed variable programming Risk aversion

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Pattern search ranking and selection algorithms for mixed variable simulation-based optimization

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2009年第3期198卷 878-890页

作者： Sriver, Todd A. Chrissis, James W. Abramson, Mark A. USAF Inst Technol Dept Operat Sci Wright Patterson AFB OH 45433 USA Headquarters US Air Forces Europe Anal Lessons Learned Div Ramstein Air Base APO AE 09094 USA Boeing Co Seattle WA 98124 USA

The class of generalized pattern search (GPS) algorithms for mixed variable optimization is extended to problems with stochastic objective functions. Because random noise in the objective function makes it more difficult to compare trial points and ascertain which points are truly better than others, replications are needed to generate sufficient statistical power to draw conclusions. Rather than comparing pairs of points, the approach taken here augments pattern search with a ranking and selection (R&S) procedure, which allows for comparing many function values simultaneously. Asymptotic convergence for the algorithm is established, numerical issues are discussed, and performance of the algorithm is studied on a set of test problems. Published by Elsevier B.V.

关键词： Pattern search algorithms mixed variable programming Nonlinear programming Simulation Ranking and selection

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An algorithm model for mixed variable programming

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SIAM JOURNAL ON OPTIMIZATION 2005年第4期15卷 1057-1084页

作者： Lucidi, S Piccialli, V Sciandrone, M Univ Roma La Sapienza Dipartimento Informat & Sistemist A Ruberti I-00185 Rome Italy CNR Ist Anal Sistemi & Informat I-00185 Rome Italy

In this paper we consider a particular class of nonlinear optimization problems involving both continuous and discrete variables. The distinguishing feature of this class of nonlinear mixed variable optimization problems is that the structure and the number of variables of the problem depend on the values of some discrete variables. In particular, we de. ne a general algorithm model for the solution of this class of problems, that draws inspiration from the approach recently proposed by Audet and Dennis [SIAM J. Optim., 11 (2001), pp. 573-594], and is based on the strategy of combining in a suitable way a local search with respect to the continuous variables and a local search with respect to the discrete variables. We prove global convergence of the algorithm model without specifying the local continuous search, but only identifying some reasonable requirements. Moreover, we de. ne a particular derivative-free algorithm for solving mixed variable programming problems where the continuous variables are linearly constrained and derivative information is not available. Finally, we report numerical results obtained by the proposed algorithm in solving a real optimal design problem. These results show the effectiveness of the approach.

关键词： nonlinear optimization mixed variable programming derivative-free methods

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Optimal sensor placement for enhancing sensitivity to change in stiffness for structural health monitoring

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OPTIMIZATION AND ENGINEERING 2008年第2期9卷 119-142页

作者： Beal, Josh M. Shukla, Amit Brezhneva, Olga A. Abramson, Mark A. Miami Univ Sch Engn & Appl Sci Dept Mech & Mfg Engn Oxford OH 45056 USA Ohio Univ Dept Math Athens OH 45701 USA Miami Univ Dept Math & Stat Oxford OH 45056 USA USAF Inst Technol Dept Math & Stat Wright Patterson AFB OH 45433 USA

This paper focuses on optimal sensor placement for structural health monitoring (SHM), in which the goal is to find an optimal configuration of sensors that will best predict structural damage. The problem is formulated as a bound constrained mixed variable programming (MVP) problem, in which the discrete variables are categorical;i.e., they may only take on values from a pre-defined list. The problem is particularly challenging because the objective function is computationally expensive to evaluate and first-order derivatives may not be available. The problem is solved numerically using the generalized mixed variable pattern search (MVPS) algorithm. Some new theoretical convergence results are proved, and numerical results are presented, which show the potential of our approach.

关键词： mixed variable programming optimal sensor placement structural health monitoring

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Pattern search algorithms for mixed variable programming

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SIAM JOURNAL ON OPTIMIZATION 2001年第3期11卷 573-594页

作者： Audet, C Dennis, JE Ecole Polytech GERAD Dept Math & Genie Ind Montreal PQ H3C 3A7 Canada Rice Univ Dept Computat & Appl Math Houston TX 77005 USA

Many engineering optimization problems involve a special kind of discrete variable that can be represented by a number, but this representation has no significance. Such variables arise when a decision involves some situation like a choice from an unordered list of categories. This has two implications: The standard approach of solving problems with continuous relaxations of discrete variables is not available, and the notion of local optimality must be defined through a user-specified set of neighboring points. We present a class of direct search algorithms to provide limit points that satisfy some appropriate necessary conditions for local optimality for such problems. We give a more expensive version of the algorithm that guarantees additional necessary optimality conditions. A small example illustrates the differences between the two versions. A real thermal insulation system design problem illustrates the efficacy of the user controls for this class of algorithms.

关键词： pattern search algorithm convergence analysis bound constrained optimization mixed variable programming derivative-free optimization

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A derivative-based algorithm for a particular class of mixed variable optimization problems

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OPTIMIZATION METHODS & SOFTWARE 2004年第3-4期19卷 371-387页

作者： Lucidi, S Piccialli, V Univ Roma La Sapienza Dipartimento Informat & Sistemist I-00185 Rome Italy

In this article, we consider a particular class of nonlinear mixed variable optimization problems where the structure and the number of variables of the problem depend on the values of some discrete variables. The peculiarity of this class is that, for fixed values of the integer variables, the corresponding continuous optimization problem contains no constraints and a large number of variables. For such a class of problems we propose two minimization algorithms and prove their global convergence properties.

关键词： mixed variable programming large-scale unconstrained optimization nonlinear programing

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mixed variable optimization of a load-bearing thermal insulation system using a filter pattern search algorithm

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OPTIMIZATION AND ENGINEERING 2004年第2期5卷 157-177页

作者： Abramson, MA USAF Inst Technol Dept Math & Stat Wright Patterson AFB OH 45433 USA

This paper describes the optimization of a load-bearing thermal insulation system characterized by hot and cold surfaces with a series of heat intercepts and insulators between them. The optimization problem is represented as a mixed variable programming (MVP) problem with nonlinear constraints, in which the objective is to minimize the power required to maintain the heat intercepts at fixed temperatures so that one surface is kept sufficiently cold. MVP problems are more general than mixed integer nonlinear programming (MINLP) problems in that the discrete variables are categorical;i.e., they must always take on values from a predefined enumerable set or list. Thus, traditional approaches that use branch and bound techniques cannot be applied. In a previous paper, a linearly constrained version of this problem was solved numerically using the Audet-Dennis generalized pattern search (GPS) method for MVP problems. However, this algorithm may not work for problems with general nonlinear constraints. A new algorithm that extends that of Audet and Dennis by incorporating a filter to handle nonlinear constraints makes it possible to solve the more general problem. Additional nonlinear constraints on stress, mass, and thermal contraction are added to that of the previous work in an effort to find a more realistic feasible design. Several computational experiments show a substantial improvement in power required to maintain the system, as compared to the previous literature. The addition of the new constraints leads to a very different design without significantly changing the power required. The results demonstrate that the new algorithm can be applied to a very broad class of optimization problems, for which no previous algorithm with provable convergence results could be applied.

关键词： mixed variable programming categorical variables filter algorithm pattern search algorithm thermal insulation heat intercepts nonlinear constraints

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A derivative-based algorithm for a particular class of mixed variable optimization problems

A derivative-based algorithm for a particular class of mixed...

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1st International Conference on Optimization Methods and Software (OMS2002)

作者： Lucidi, S Piccialli, V Univ Roma La Sapienza Dipartimento Informat & Sistemist I-00185 Rome Italy

关键词： mixed variable programming large-scale unconstrained optimization nonlinear programing

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