Class Of Multi-objective Optimization Problems Research Articles

A line search technique for a class of multi-objective optimization problems using subgradient

A line search technique for a class of multi-objective optimization problems using subgradient

Duality Theorems for (ρ,ψ,d)-Quasiinvex Multiobjective Optimization Problems with Interval-Valued Components

The present paper deals with a duality study associated with a new class of multiobjective optimization problems that include the interval-valued components of the ratio vector. More precisely, by using the new notion of (ρ,ψ,d)-quasiinvexity associated with an interval-valued multiple-integral functional, we formulate and prove weak, strong, and converse duality results for the considered class of variational control problems.

One-exact approximate Pareto sets

Papadimitriou and Yannakakis (Proceedings of the 41st annual IEEE symposium on the Foundations of Computer Science (FOCS), pp 86–92, 2000) show that the polynomial-time solvability of a certain auxiliary problem determines the class of multiobjective optimization problems that admit a polynomial-time computable (1+varepsilon , dots , 1+varepsilon )-approximate Pareto set (also called an varepsilon -Pareto set). Similarly, in this article, we characterize the class of multiobjective optimization problems having a polynomial-time computable approximate varepsilon -Pareto set that is exact in one objective by the efficient solvability of an appropriate auxiliary problem. This class includes important problems such as multiobjective shortest path and spanning tree, and the approximation guarantee we provide is, in general, best possible. Furthermore, for biobjective optimization problems from this class, we provide an algorithm that computes a one-exact varepsilon -Pareto set of cardinality at most twice the cardinality of a smallest such set and show that this factor of 2 is best possible. For three or more objective functions, however, we prove that no constant-factor approximation on the cardinality of the set can be obtained efficiently.

A Multi-Objective Interpretation of Optimal Transport

This paper connects discrete optimal transport to a certain class of multi-objective optimization problems. In both settings, the decision variables can be organized into a matrix. In the multi-objective problem, the notion of Pareto efficiency is defined in terms of the objectives together with nonnegativity constraints and with equality constraints that are specified in terms of column sums. A second set of equality constraints, defined in terms of row sums, is used to single out particular points in the Pareto-efficient set which are referred to as “balanced solutions.” Examples from several fields are shown in which this solution concept appears naturally. Balanced solutions are shown to be in one-to-one correspondence with solutions of optimal transport problems. As an example of the use of alternative interpretations, the computation of solutions via regularization is discussed.

A linear programming based algorithm to solve a class of optimization problems with a multi-linear objective function and affine constraints

We present a linear programming based algorithm for a class of optimization problems with a multi-linear objective function and affine constraints. This class of optimization problems has only one objective function, but it can also be viewed as a class of multi-objective optimization problems by decomposing its objective function. The proposed algorithm exploits this idea and solves this class of optimization problems from the viewpoint of multi-objective optimization. The algorithm computes an optimal solution when the number of variables in the multi-linear objective function is two, and an approximate solution when the number of variables is greater than two. A computational study demonstrates that when available computing time is limited the algorithm significantly outperforms well-known convex programming solvers IPOPT and CVXOPT, in terms of both efficiency and solution quality. The optimization problems in this class can be reformulated as second-order cone programs, and, therefore, also be solved by second-order cone programming solvers. This is highly effective for small and medium size instances, but we demonstrate that for large size instances with two variables in the multi-linear objective function the proposed algorithm outperforms a (commercial) second-order cone programming solver.

A hybrid multi-objective Particle Swarm Optimization for scientific workflow scheduling

Now-a-days, Cloud computing is a technology which eludes provision cost while providing scalability and elasticity to accessible resources on a pay-per-use basis. To satisfy the increasing demand of the computing power to execute large scale scientific workflow applications, workflow scheduling is the main challenging issue in Infrastructure-as-a-Service (IaaS) clouds. As workflow scheduling belongs to NP-complete problem, so, meta-heuristic approaches are more preferred option. Users often specified deadline and budget constraint for scheduling these workflow applications over cloud resources. But these constraints are in conflict with each other, i.e., the cheaper resources are slow as compared to the expensive resources. Most of the existing studies try to optimize only one of the objectives, i.e., either time minimization or cost minimization under user specified Quality of Service (QoS) constraints. But due to the complexity of workflows and dynamic nature of cloud, a trade-off solution is required to make a balance between execution time and processing cost. To address these issues, this paper presents a non-dominance sort based Hybrid Particle Swarm Optimization (HPSO) algorithm to handle the workflow scheduling problem with multiple conflicting objective functions on IaaS clouds. The proposed algorithm is a hybrid of our previously proposed Budget and Deadline constrained Heterogeneous Earliest Finish Time (BDHEFT) algorithm and multi-objective PSO. The HPSO heuristic tries to optimize two conflicting objectives, namely, makespan and cost under the deadline and budget constraints. Along with these two conflicting objectives, energy consumed of created workflow schedule is also minimized. The proposed algorithm gives a set of Pareto Optimal solutions from which the user can choose the best solution. The performance of proposed heuristic is compared with state-of-art multi-objective meta-heuristics like NSGA-II, MOPSO, and ε-FDPSO. The simulation analysis substantiates that the solutions obtained with proposed heuristic deliver better convergence and uniform spacing among the solutions as compared to others. Hence it is applicable to solve a wide class of multi-objective optimization problems for scheduling scientific workflows over IaaS clouds.

Homotopy method for a class of multiobjective optimization problems with equilibrium constraints

In this paper, we present a combined homotopy interior point method for solving multiobjective programs with equilibrium constraints. Under suitable conditions, we prove the existence and convergence of a smooth homotopy path from almost any interior point to a solution of the K-K-T system. Numerical results are presented to show the effectiveness of this algorithm.

Interactive Polyhedral Outer Approximation (IPOA) strategy for general multiobjective optimization problems

We propose an interactive polyhedral outer approximation (IPOA) method to solve a broad class of multiobjective optimization problems (MOP) with, possibly, nonlinear and nondifferentiable objective and constraint functions, and with continuous or discrete decision variables. During the interactive optimization phase, the method progressively constructs a polyhedral approximation of the decision-maker’s (DM’s) unknown preference structure and a polyhedral outer-approximation of the feasible set of MOP. The piecewise linear approximation of the DM’s preferences also provides a mechanism for testing the consistency of the DM’s assessments and removing inconsistencies; it also allows post-optimality analysis. All the feasible trial solutions are non-dominated (efficient, or Pareto-optimal) so preference assessments are made in the context of non-dominated alternatives only. Upper and lower bounds on the yet unknown optimal value are produced at every iteration, allowing terminating the search prematurely at a good-enough solution and providing information about the closeness of this solution to the optimal solution. The IPOA method includes a preliminary phase in which a limited probe of the efficient set is conducted in order to find a good initial trial solution for the interactive phase. The computational requirements of the algorithm are relatively simple. The results of an extensive computational study are reported.

Application of Genetic Algorithm for Solving Multi-Objective Optimization Problems in Robust Control of Distillation Column

A genetic algorithm (GA) for the class of multi-objective optimization problems that appears in the design of robust controllers is presented in this paper. The design of a robust controller is a trade-off problem among competitive objectives such as disturbance rejection, reference tracking, stability against un-modeled dynamics, moderate control effort and so on. However, general methodologies for solving this class of design problems are not easily encountered in the literature because of the complexity of the resultant multi-objective problems. In this paper, popular adaptations of the simple GA, non-dominated sorting genetic algorithms, are used to solve robust control design problems. The structure and operators of this algorithm have been specifically developed for control design problems. The performance of the algorithm is evaluated by solving several test cases and is also compared to the standard algorithms used for the multi-objective design of robust controllers.

On Economic Equilibrium Type Problems with Applications

The existence solution for a class of economic equilibrium type problems in reflexive Banach space is considered. New results concerning the variational inequality approach to Arrow–Debreu model of economic equilibrium introduced in Naniewicz (Math Oper Res 32:436–466, 2007) are found and applied to ensure the existence of Pareto optimal solutions for a class of multiobjective optimization problems with so-called budget-like constraints. To achieve this goal, the theory of pseudo-monotone multivalued mappings combined with some fixed point technique for multivalued mappings with nonconvex values is used.

Approximating the Set of Pareto-Optimal Solutions in Both the Decision and Objective Spaces by an Estimation of Distribution Algorithm

Most existing multiobjective evolutionary algorithms aim at approximating the Pareto front (PF), which is the distribution of the Pareto-optimal solutions in the objective space. In many real-life applications, however, a good approximation to the Pareto set (PS), which is the distribution of the Pareto-optimal solutions in the decision space, is also required by a decision maker. This paper considers a class of multiobjective optimization problems (MOPs), in which the dimensionalities of the PS and the PF manifolds are different so that a good approximation to the PF might not approximate the PS very well. It proposes a probabilistic model-based multiobjective evolutionary algorithm, called MMEA, for approximating the PS and the PF simultaneously for an MOP in this class. In the modeling phase of MMEA, the population is clustered into a number of subpopulations based on their distribution in the objective space, the principal component analysis technique is used to estimate the dimensionality of the PS manifold in each subpopulation, and then a probabilistic model is built for modeling the distribution of the Pareto-optimal solutions in the decision space. Such a modeling procedure could promote the population diversity in both the decision and objective spaces. MMEA is compared with three other methods, KP1, Omni-Optimizer and RM-MEDA, on a set of test instances, five of which are proposed in this paper. The experimental results clearly suggest that, overall, MMEA performs significantly better than the three compared algorithms in approximating both the PS and the PF.

Pareto optimality and Walrasian equilibria

The paper concerns the study of a class of convex, constrained multiobjective optimization problems from the viewpoint of the existence issues. The main feature of the presented approach is that the classical qualification condition requiring the existence of interior points in the effective domains of functions under consideration does not hold. A variant of duality theory for multiobjective optimization problems based on the Fenchel theorem is formulated. Next, by using very recent results on the Walrasian general equilibrium model of economy obtained in Naniewicz [Z. Naniewicz, Pseudo-monotonicity and economic equilibrium problem in reflexive Banach space, Math. Oper. Res. 32 (2007) 436–466] the conditions ensuring the existence of Pareto optimal solutions for the class of multiobjective optimization problems are established. The concept of the proper efficiency is used as the solution notion. Finally, a new version of the second fundamental theorem of welfare economics is presented.

Variable Programming: A Generalized Minimax Problem. Part I: Models and Theory

In this two-part series of papers, a new generalized minimax optimization model, termed variable programming (VP), is developed to solve dynamically a class of multi-objective optimization problems with non-decomposable structure. It is demonstrated that such type of problems is more general than existing optimization models. In this part, the VP model is proposed first, and the relationship between variable programming and the general constrained nonlinear programming is established. To illustrate its practicality, problems on investment and the low-side-lobe conformal antenna array pattern synthesis to which VP can be appropriately applied are discussed for substantiation. Then, theoretical underpinnings of the VP problems are established. Difficulties in dealing with the VP problems are discussed. With some mild assumptions, the necessary conditions for the unconstrained VP problems with arbitrary and specific activated feasible sets are derived respectively. The necessary conditions for the corresponding constrained VP problems with the mild hypotheses are also examined. Whilst discussion in this part is concentrated on the formulation of the VP model and its theoretical underpinnings, construction of solution algorithms is discussed in Part II.

On the approximate tradeoff for bicriteria batching and parallel machine scheduling problems

We consider multiobjective scheduling problems, i.e. scheduling problems that are evaluated with respect to many cost criteria, and we are interested in determining a trade-off (Pareto curve) among these criteria. We study two types of bicriteria scheduling problems: single-machine batching problems and parallel machine scheduling problems. Instead of proceeding in a problem-by-problem basis, we identify a class of multiobjective optimization problems possessing a fully polynomial time approximation scheme (FPTAS) for computing an ε-approximate Pareto curve. This class contains a set of problems whose Pareto curve can be computed via a simple pseudo-polynomial dynamic program for which the objective and transition functions satisfy some, easy to verify, arithmetical conditions. Our study is based on a recent work of Woeginger (Electronic Colloquium on Computational Complexity, Report 84 (short version appeared in SODA’99, pp. 820–829)) for the single criteria optimization ex-benevolent problems. We show how our general result can be applied to the considered scheduling problems.

MRCD: a genetic algorithm for multiobjective robust control design

A genetic algorithm (GA) for the class of multiobjective optimization problems that appears in the design of robust controllers is presented in this paper. The design of a robust controller is a trade-off problem among competitive objectives such as disturbance rejection, reference tracking, stability against unmodeled dynamics, moderate control effort and so on. However, general methodologies for solving this class of design problems are not easily encountered in the literature because of the complexity of the resultant multiobjective problems. In this paper, a recently developed class of GAs, multiobjective GAs, are used to solve robust control design problems. Here, a new algorithm, called multiobjective robust control design, has been proposed. The structure and operators of this algorithm have been specifically developed for control design problems. The performace of the algorithm is evaluated by solving several test cases and is also compared to the standard algorithms used for the multiobjective design of robust controllers.

An Interactive Optimization Method for a Water Resource Problem with Multiple Objectives

Some class of multiobjective optimization problems may be effectively solved by interactive programming methods. First, in this paper, a new type of interactive optimization technique is suggested. The most prominent feature of the method is in the fact that the method requires only the partial informations about the preference attitude of the decision maker which is most easily extracted from him. Next, the method is applied to a water quality problem in the Yodo River basin in Japan. The stated problem includes several kinds of conflicts and may be formulated as a multiobjective optimization problem. The obtained numerical results prove the effectiveness of the method.