Quasi-Newton methods for solving multiobjective optimization

This paper presents a quasi-Newton-type algorithm for nonconvex multiobjective optimization. In this algorithm, the iterations are repeated until termination conditions are met, which is when a suitable descent direction cannot be found anymore. Under suitable assumptions, global convergence is established.     

On solving multiobjective bin packing problems using evolutionary particle swarm optimization

The bin packing problem is widely found in applications such as loading of tractor trailer trucks, cargo airplanes and ships, where a balanced load provides better fuel efficiency and safer ride. In these applications, there are often conflicting criteria to be satisfied, i.e., to minimize the bins used and to balance the load of each bin, subject to a number of practical constraints. Unlike existing studies that only consider the issue of minimum bins, a multiobjective two-dimensional mathematical model for bin packing problems with multiple constraints (MOBPP-2D) is formulated in this paper. To solve MOBPP-2D problems, a multiobjective evolutionary particle swarm optimization algorithm (MOEPSO) is proposed. Without the need of combining both objectives into a composite scalar weighting function, MOEPSO incorporates the concept of Pareto’s optimality to evolve a family of solutions along the trade-off surface. Extensive numerical investigations are performed on various test instances, and their performances are compared both quantitatively and statistically with other optimization methods to illustrate the effectiveness and efficiency of MOEPSO in solving multiobjective bin packing problems.     

A novel evolution strategy for multiobjective optimization problem

Recent literatures have suggested that multiobjective evolutionary algorithms (MOEAs) can serve as a more exploratory and effective tool in solving multiobjective optimization problems (MOPs) than traditional optimizers. In order to contain a good approximation of Pareto optimal set with wide diversity associated with the inherent characters and variability of MOPs, this paper proposes a new evolutionary approach—(μ, λ) multiobjective evolution strategy ((μ, λ)-MOES). Following the highlight of how to balance proximity and diversity of individuals in exploration and exploitation stages respectively, some cooperative techniques are devised. Firstly, a novel combinatorial exploration operator that develops strong points from Gaussian mutation of proximity exploration and from Cauchy mutation of diversity preservation is elaborately designed. Additionally, we employ a complete nondominance selection so as to ensure maximal pressure for proximity exploitation while a fitness assignment determined by dominance and population diversity information is simultaneous used to ensure maximal diversity preservation. Moreover, a dynamic external archive is introduced to store elitist individuals as well as relatively better individuals and exchange information with the current population when performing archive increase scheme and archive decrease scheme. By graphical presentation and examination of selected performance metrics on three prominent benchmark test functions, (μ, λ)-MOES is found to outperform SPEA-II to some extent in terms of finding a near-optimal, well-extended and uniformly diversified Pareto optimal front.     

A survey of recent developments in multiobjective optimization

Multiobjective Optimization (MO) has many applications in such fields as the Internet, finance, biomedicine, management science, game theory and engineering. However, solving MO problems is not an easy task. Searching for all Pareto optimal solutions is expensive and a time consuming process because there are usually exponentially large (or infinite) Pareto optimal solutions. Even for simple problems determining whether a point belongs to the Pareto set is -hard. In this paper, we discuss recent developments in MO. These include optimality conditions, applications, global optimization techniques, the new concept of epsilon Pareto optimal solution, and heuristics.     

Explicit gradient information in multiobjective optimization

This work presents an algorithm that converges to points that satisfy a first-order necessary condition of weakly Pareto solutions of multiobjective optimization problems. Hints on how to include second-order information are given. Preliminary numerical results are encouraging.     

Some kind of Pareto stationarity for multiobjective problems with equilibrium constraints

ABSTRACT

In this paper, we derive some optimality and stationarity conditions for a multiobjective problem with equilibrium constraints (MOPEC). In particular, under a generalized Guignard constraint qualification, we show that any locally Pareto optimal solution of MOPEC must satisfy the strong Pareto Kuhn-Tucker optimality conditions. We also prove that the generalized Guignard constraint qualification is the weakest constraint qualification for the strong Pareto Kuhn-Tucker optimality. Furthermore, under certain convexity or generalized convexity assumptions, we show that the strong Pareto Kuhn-Tucker optimality conditions are also sufficient for several popular locally Pareto-type optimality conditions for MOPEC.     

A gap between multiobjective optimization and scalar optimization

We give an example to illustrate a gap between multiobjective optimization and single-objective optimization, which solves a problem proposed in Ref. 1.     

The Pareto fitness genetic algorithm: Test function study

Evolutionary algorithms have shown some success in solving multiobjective optimization problems. The methods of fitness assignment are mainly based on the information about the dominance relation between individuals. We propose a Pareto fitness genetic algorithm (PFGA) in which we introduce a modified ranking procedure and a promising way of sharing; a new fitness function based on the rank of the individual and its density value is designed. This is considered as our main contribution. The performance of our algorithm is evaluated on six multiobjective benchmarks with different Pareto front features. Computational results (quality of the approximation of the Pareto optimal set and the number of fitness function evaluations) proving its efficiency are reported.     

A new multiobjective simulated annealing algorithm

A new multiobjective simulated annealing algorithm for continuous optimization problems is presented. The algorithm has an adaptive cooling schedule and uses a population of fitness functions to accurately generate the Pareto front. Whenever an improvement with a fitness function is encountered, the trial point is accepted, and the temperature parameters associated with the improving fitness functions are cooled. Beside well known linear fitness functions, special elliptic and ellipsoidal fitness functions, suitable for the generation on non-convex fronts, are presented. The effectiveness of the algorithm is shown through five test problems. The parametric study presented shows that more fitness functions as well as more iteration gives more non-dominated points closer to the actual front. The study also compares the linear and elliptic fitness functions. The success of the algorithm is also demonstrated by comparing the quality metrics obtained to those obtained for a well-known evolutionary multiobjective algorithm.     

An approach for solving a fuzzy multiobjective programming problem

In this paper we present a new approach, based on the Nearest Interval Approximation Operator, for dealing with a multiobjective programming problem with fuzzy-valued objective functions.     

A hybrid method for solving multi-objective global optimization problems

Real optimization problems often involve not one, but multiple objectives, usually in conflict. In single-objective optimization there exists a global optimum, while in the multi-objective case no optimal solution is clearly defined but rather a set of optimums, which constitute the so called Pareto-optimal front. Thus, the goal of multi-objective strategies is to generate a set of non-dominated solutions as an approximation to this front. However, most problems of this kind cannot be solved exactly because they have very large and highly complex search spaces. The objective of this work is to compare the performance of a new hybrid method here proposed, with several well-known multi-objective evolutionary algorithms (MOEA). The main attraction of these methods is the integration of selection and diversity maintenance. Since it is very difficult to describe exactly what a good approximation is in terms of a number of criteria, the performance is quantified with adequate metrics that evaluate the proximity to the global Pareto-front. In addition, this work is also one of the few empirical studies that solves three-objective optimization problems using the concept of global Pareto-optimality.     

Weak Pareto optimality of multiobjective problems in a locally convex linear topological space

Several corrections to Ref. 1 are pointed out.     

Weak Pareto optimality of multiobjective problems in a locally convex linear topological space

In this study, the Dubovitskii-Milyutin type optimization theory is extended to multiobjective programs in a locally convex linear topological space, producing necessary conditions for a weak Pareto optimum. In the case of an ordinary multiobjective convex program, generalized Kuhn-Tucker conditions by a subdifferential formula are necessary and sufficient for a weak Pareto optimum.The author is grateful for the useful suggestions and comments of Professor N. Furukawa and the referee.     

A clustering procedure for reducing the number of representative solutions in the Pareto Front of multiobjective optimization problems

In many multiobjective optimization problems, the Pareto Fronts and Sets contain a large number of solutions and this makes it difficult for the decision maker to identify the preferred ones. A possible way to alleviate this difficulty is to present to the decision maker a subset of a small number of solutions representatives of the Pareto Front characteristics.     

On the choice of special Pareto points

Abstract

We propose two strategies for choosing Pareto solutions of constrained multiobjective optimization problems. The first one, for general problems, furnishes balanced optima, i.e. feasible points that, in some sense, have the closest image to the vector whose coordinates are the objective components infima. It consists of solving a single scalar-valued problem, whose objective requires the use of a monotonic function which can be chosen within a large class of functions. The second one, for practical problems for which there is a preference among the objective’s components to be minimized, gives us points that satisfy this order criterion. The procedure requires the sequential minimization of all these functions. We also study other special Pareto solutions, the sub-balanced points, which are a generalization of the balanced optima.     

Necessary conditions of Pareto optimality for multiobjective optimal control problems under constraints

This paper studies multiobjective optimal control problems in presence of constraints in the discrete time framework. Both the finite- and infinite-horizon settings are considered. The paper provides necessary conditions of Pareto optimality under lighter smoothness assumptions compared to the previously obtained results. These conditions are given in the form of weak and strong Pontryagin principles which generalize the existing ones. To obtain some of these results, we provide new multiplier rules for multiobjective static optimization problems and new Pontryagin principles for the finite horizon multiobjective optimal control problems.     

Approximate necessary conditions for locally weak Pareto optimality

Necessary conditions for a given pointx ⁰ to be a locally weak solution to the Pareto minimization problem of a vector-valued functionF=(f ₁,...,f _m ),F:XR ^m,XR ^m, are presented. As noted in Ref. 1, the classical necessary condition-conv {Df ₁(x ⁰)|i=1,...,m}T ^*(X, x ⁰) need not hold when the contingent coneT is used. We have proven, however, that a properly adjusted approximate version of this classical condition always holds. Strangely enough, the approximation form>2 must be weaker than form=2.The authors would like to thank the anonymous referee for the suggestions which led to an improved presentation of the paper.     

A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions

In a general Hilbert framework, we consider continuous gradient-like dynamical systems for constrained multiobjective optimization involving nonsmooth convex objective functions. Based on the Yosida regularization of the subdifferential operators involved in the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and the convergence of trajectories to weak Pareto minima. This approach provides a dynamical endogenous weighting of the objective functions, a key property for applications in cooperative games, inverse problems, and numerical multiobjective optimization.     

Asymptotic convergence of a simulated annealing algorithm for multiobjective optimization problems

In this paper we consider a simulated annealing algorithm for multiobjective optimization problems. With a suitable choice of the acceptance probabilities, the algorithm is shown to converge asymptotically, that is, the Markov chain that describes the algorithm converges with probability one to the Pareto optimal set.     

A real coded genetic algorithm for solving integer and mixed integer optimization problems

In this paper, a real coded genetic algorithm named MI-LXPM is proposed for solving integer and mixed integer constrained optimization problems. The proposed algorithm is a suitably modified and extended version of the real coded genetic algorithm, LXPM, of Deep and Thakur [K. Deep, M. Thakur, A new crossover operator for real coded genetic algorithms, Applied Mathematics and Computation 188 (2007) 895-912; K. Deep, M. Thakur, A new mutation operator for real coded genetic algorithms, Applied Mathematics and Computation 193 (2007) 211-230]. The algorithm incorporates a special truncation procedure to handle integer restrictions on decision variables along with a parameter free penalty approach for handling constraints. Performance of the algorithm is tested on a set of twenty test problems selected from different sources in literature, and compared with the performance of an earlier application of genetic algorithm and also with random search based algorithm, RST2ANU, incorporating annealing concept. The proposed MI-LXPM outperforms both the algorithms in most of the cases which are considered.