LMI Approach to Robust Model Predictive Control

This paper introduces a new approach to robust model predictive control (MPC) based on conservative approximations to semi-infinite optimization using linear matrix inequalities (LMIs). The method applies to problems with convex quadratic costs, linear and convex quadratic constraints, and linear predictive models with bounded uncertainty. If the MPC optimization problem is feasible at the initial control step (the first application of the MPC optimization), it is shown that the MPC optimization problems will be feasible at all future time steps and that the controlled system will be closed-loop stable. The method is illustrated with a solenoid control example. The authors thank the anonymous reviewers for suggestions that improved the presentation of this work. The work was supported in part by the EPRI/DoD Complex Interactive Networks/Systems Initiative under Contract EPRI-W08333-05 and by the US Army Research Office Contract DAAD19-01-1-0485.     

Robustness to uncertain optimization using scalarization techniques and relations to multiobjective optimization

In this paper, we propose two kinds of robustness concepts by virtue of the scalarization techniques (Benson’s method and elastic constraint method) in multiobjective optimization, which can be characterized as special cases of a general non-linear scalarizing approach. Moreover, we introduce both constrained and unconstrained multiobjective optimization problems and discuss their relations to scalar robust optimization problems. Particularly, optimal solutions of scalar robust optimization problems are weakly efficient solutions for the unconstrained multiobjective optimization problem, and these solutions are efficient under uniqueness assumptions. Two examples are employed to illustrate those results. Finally, the connections between robustness concepts and risk measures in investment decision problems are also revealed.     

Optimality and Duality for a Class of Nondifferentiable Multiobjective Fractional Programming Problems

In this paper, we consider a class of nondifferentiable multiobjective fractional programs in which each component of the objective function contains a term involving the support function of a compact convex set. We establish necessary and sufficient optimality conditions and duality results for weakly efficient solutions of nondifferentiable multiobjective fractional programming problems. This work was supported by Grant R01-2003-000-10825-0 from the Basic Research Program of KOSEF.     

Sharp Minima for Multiobjective Optimization in Banach Spaces

We study sharp minima for multiobjective optimization problems. In terms of the Mordukhovich coderivative and the normal cone, we present sufficient and or necessary conditions for existence of such sharp minima, some of which are new even in the single objective setting.This research was supported by a Central Research Grant of The Hong Kong Polytechnic University (Grant No. G-T 507). Research of the first author was also supported by the National Natural Science Foundation of PR China (Grant No. 10361008) and the Natural Science Foundation of Yunnan Province, China (Grant No. 2003A002M).     

Duality for Set-Valued Multiobjective Optimization Problems,Part 1: Mathematical Programming

The duality of multiobjective problems is studied with the help of the apparatus of conjugate set-valued mappings introduced by the author. In this paper (Part 1), a duality theory is developed for set-valued mappings, which is then used to derive dual relations for some general multiobjective optimization problems which include convex programming and optimal control problems. Using this result, in the companion paper (Part 2), duality theorems are proved for multiobjective quasilinear and linear optimal control problems. The theory is applied to get dual relations for some multiobjective optimal control problem.     

An empirical study on similarity-based mating for evolutionary multiobjective combinatorial optimization

We have already proposed a similarity-based mating scheme to recombine extreme and similar parents for evolutionary multiobjective optimization. In this paper, we examine the effect of the similarity-based mating scheme on the performance of evolutionary multiobjective optimization (EMO) algorithms. First we examine which is better between recombining similar or dissimilar parents. Next we examine the effect of biasing selection probabilities toward extreme solutions that are dissimilar from other solutions in each population. Then we examine the effect of dynamically changing the strength of this bias during the execution of EMO algorithms. Computational experiments are performed on a wide variety of test problems for multiobjective combinatorial optimization. Experimental results show that the performance of EMO algorithms can be improved by the similarity-based mating scheme for many test problems.     

A novel elitist multiobjective optimization algorithm: Multiobjective extremal optimization

Recently, a general-purpose local-search heuristic method called extremal optimization (EO) has been successfully applied to some NP-hard combinatorial optimization problems. This paper presents an investigation on EO with its application in numerical multiobjective optimization and proposes a new novel elitist (1 + λ) multiobjective algorithm, called multiobjective extremal optimization (MOEO). In order to extend EO to solve the multiobjective optimization problems, the Pareto dominance strategy is introduced to the fitness assignment of the proposed approach. We also present a new hybrid mutation operator that enhances the exploratory capabilities of our algorithm. The proposed approach is validated using five popular benchmark functions. The simulation results indicate that the proposed approach is highly competitive with the state-of-the-art multiobjective evolutionary algorithms. Thus MOEO can be considered a good alternative to solve numerical multiobjective optimization problems.     

Generalized Homotopy Approach to Multiobjective Optimization

This paper proposes a new generalized homotopy algorithm for the solution of multiobjective optimization problems with equality constraints. We consider the set of Pareto candidates as a differentiable manifold and construct a local chart which is fitted to the local geometry of this Pareto manifold. New Pareto candidates are generated by evaluating the local chart numerically. The method is capable of solving multiobjective optimization problems with an arbitrary number k of objectives, makes it possible to generate all types of Pareto optimal solutions, and is able to produce a homogeneous discretization of the Pareto set. The paper gives a necessary and sufficient condition for the set of Pareto candidates to form a (k-1)-dimensional differentiable manifold, provides the numerical details of the proposed algorithm, and applies the method to two multiobjective sample problems.     

求解群体多目标最优化问题的联合有效数法

群体多目标最优化是群体决策和多目标最优化相交叉的一个边缘研究领域,其主要特点是对由多个决策者提供的具多个目标的最优化问题,进行定量和定性相结合的方案选优或决策排序.因此,它的理论和方法在现代社会的重大决策中有着广阔的应用前景.     

Hard uncertainties in multiobjective layout optimization of photovoltaic power plants

This paper describes how to treat hard uncertainties defined by so-called uncertainty maps in multiobjective optimization problems. For the uncertainty map being set-valued, a Taylor formula is shown under appropriate assumptions. The hard uncertainties are modeled using parametric set optimization problems for which a scalarization result is given. The presented new approach for the solution of multiobjective optimization problems with hard uncertainties is then applied to the layout optimization of photovoltaic power plants. Since good weather forecasts are difficult to obtain for future years, weather data are really hard uncertainties arising in the planning process. Numerical results are presented for a real-world problem on the Galapagos island Isabela.     

Solving geometric programs using GRG: Results and comparisons

This paper describes the performance of a general-purpose GRG code for nonlinear programming in solving geometric programs. The main conclusions drawn from the experiments reported are: (i) GRG competes well with special-purpose geometric programming codes in solving geometric programs; and (ii) standard time, as defined by Colville, is an inadequate means of compensating for different computing environments while comparing optimization algorithms.This research was partially supported by the Office of Naval Research under Contracts Nos. N00014-75-C-0267 and N00014-75-C-0865, the US Energy Research and Development Administration, Contract No. E(04-3)-326 PA-18, and the National Science Foundation, Grant No. DCR75-04544 at Stanford University; and by the Office of Naval Research under Contract No. N00014-75-C-0240, and the National Science Foundation, Grant No. SOC74-23808, at Case Western Reserve University.     

Numerical experiments with the LANCELOT package (release A) for large-scale nonlinear optimization

In this paper, we describe the algorithmic options of Release A of LANCELOT, a Fortran package for large-scale nonlinear optimization. We then present the results of intensitive numberical tests and discuss the relative merits of the options. The experiments described involve both academic and applied problems. Finally, we propose conclusion, both specific to LANCELOT and of more general scope. This research was supported in part by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Air Force Office of Scientific Research under Contract No F49620-91-C-0079     

Nonscalarized multiobjective global optimization

A new approach to multiobjective optimization is presented which is made possible due to our ability to obtain full global optimal solutions. A distinctive feature of this approach is that a vector cost function is nonscalarized. The method provides a means for the solution of vector optimization problems with nonreconcilable objectives.This work was supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A3492.     

Equality Constraints in Multiobjective Robust Design Optimization: Decision Making Problem

Robust design optimization (RDO) problems can generally be formulated by incorporating uncertainty into the corresponding deterministic problems. In this context, a careful formulation of deterministic equality constraints into the robust domain is necessary to avoid infeasible designs under uncertain conditions. The challenge of formulating equality constraints is compounded in multiobjective RDO problems. Modeling the tradeoffs between the mean of the performance and the variation of the performance for each design objective in a multiobjective RDO problem is itself a complex task. A judicious formulation of equality constraints adds to this complexity because additional tradeoffs are introduced between constraint satisfaction under uncertainty and multiobjective performance. Equality constraints under uncertainty in multiobjective problems can therefore pose a complicated decision making problem. In this paper, we provide a new problem formulation that can be used as an effective multiobjective decision making tool, with emphasis on equality constraints. We present two numerical examples to illustrate our theoretical developments.     

Duality for multiobjective optimization problems with convex objective functions and D.C. constraints

In this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many D.C. constraints. In order to do this, we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real-valued convex functions. For a family of multiobjective problems associated to the initial one we determine then, by means of the scalar duality results, their multiobjective dual problems. Finally, we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints.     

Multiobjective Search Algorithm with Subdivision Technique

This paper presents a multiobjective search algorithm with subdivision technique (MOSAST) for the global solution of multiobjective constrained optimization problems with possibly noncontinuous objective or constraint functions. This method is based on a random search method and a new version of the Graef-Younes algorithm and it uses a subdivision technique. Numerical results are given for bicriterial test problems.     

Book Notices

Finding Pareto-minimum vectors among r given vectors, each of dimension m, is a fundamental problem in multiobjective optimization problems or multiple-criteria decision-making problems. Corley and Moon (Ref. 1) have given an algorithm for finding all the Pareto-minimum paths of a multiobjective network optimization problem from the initial node to any other node. It uses another algorithm by Corley and Moon, which actually computes the Pareto-minimum vectors. We observed that the latter algorithm is incorrect. In this note, we correct the algorithm for computing Pareto-minimum vectors and present a modified algorithm.     

Evaluation of the multiobjective ant colony algorithm performances on biobjective quadratic assignment problems

The difficulty of resolving the multiobjective combinatorial optimization problems with traditional methods has directed researchers to investigate new approaches which perform better. In recent years some algorithms based on ant colony optimization (ACO) metaheuristic have been suggested to solve these multiobjective problems. In this study these algorithms have been reported and programmed both to solve the biobjective quadratic assignment problem (BiQAP) instances and to evaluate the performances of these algorithms. The robust parameter sets for each 12 multiobjective ant colony optimization (MOACO) algorithms have been calculated and BiQAP instances in the literature have been solved within these parameter sets. The performances of the algorithms have been evaluated by comparing the Pareto fronts obtained from these algorithms. In the evaluation step, a multi significance test is used in a non hierarchical structure, and a performance metric (P metric) essential for this test is introduced. Through this study, decision makers will be able to put in the biobjective algorithms in an order according to the priority values calculated from the algorithms’ Pareto fronts. Moreover, this is the first time that MOACO algorithms have been compared by solving BiQAPs.     

Multiobjective GRASP with Path Relinking

In this paper we review and propose different adaptations of the GRASP metaheuristic to solve multiobjective combinatorial optimization problems. In particular, we describe several alternatives to specialize the construction and improvement components of GRASP when two or more objectives are considered. GRASP has been successfully coupled with Path Relinking for single-objective optimization. Moreover, we propose different hybridizations of GRASP and Path Relinking for multiobjective optimization. We apply the proposed GRASP with Path Relinking variants to two combinatorial optimization problems, the biobjective orienteering problem and the biobjective path dissimilarity problem. We report on empirical tests with 70 instances and 30 algorithms, that show that the proposed heuristics are competitive with the state-of-the-art methods for these problems.     

First and Second-Order Optimality Conditions for Convex Composite Multiobjective Optimization

Multiobjective optimization is a useful mathematical model in order to investigate real-world problems with conflicting objectives, arising from economics, engineering, and human decision making. In this paper, a convex composite multiobjective optimization problem, subject to a closed convex constraint set, is studied. New first-order optimality conditions for a weakly efficient solution of the convex composite multiobjective optimization problem are established via scalarization. These conditions are then extended to derive second-order optimality conditions.