Non-linear Multi-objective Optimization with Probabilistic Branch and Bound

The cross-entropy method in multi-objective optimisation: An assessment

Multiobjective optimization (MO) is attracting much attention of researchers in the last years. This is because most of the currently addressed optimization problems need to optimize multiple conflicting objective functions simultaneously. In this work, we propose a new algorithm called Multiobjective Artificial Bee Colony with Differential Evolution (MO-ABC/DE) for solving a set of unconstrained multiobjective optimization problems. The MO-ABC/DE algorithm is a new hybrid approach that combines the collective intelligence of the honey bee swarms (Artificial Bee Colony - ABC) with the properties of the Differential Evolution (DE). To analyse the performance of our metaheuristics we solve ten unconstrained multiobjective problems defined for the CEC 2009 Special Session on “Performance Assessment of Constrained / Bound Constrained Multiobjective Optimization Algorithms”. These test problems optimize two or three objective functions and present different properties of separability, modality, and geometry in their Pareto fronts. As we will see, the MO-ABC/DE algorithm works well on most test problems, proving to be an important tool to solve multiobjective unconstrained optimization problems.

Algorithms for multi-objective optimization problems are designed to generate a single Pareto optimum (non-dominated solution) or a set of Pareto optima that reflect the preferences of the decision-maker. If a set of Pareto optima are generated, then it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optima using an unbiased technique of filtering solutions. This suggests the need for an efficient selection procedure to identify such a preferred subset that reflects the preferences of the decision-maker with respect to the objective functions. Selection procedures typically use a value function or a scalarizing function to express preferences among objective functions. This paper introduces and analyzes the Greedy Reduction (GR) algorithm for obtaining subsets of Pareto optima from large solution sets in multi-objective optimization. Selection of these subsets is based on maximizing a scalarizing function of the vector of percentile ordinal rankings of the Pareto optima within the larger set. A proof of optimality of the GR algorithm that relies on the non-dominated property of the vector of percentile ordinal rankings is provided. The GR algorithm executes in linear time in the worst case. The GR algorithm is illustrated on sets of Pareto optima obtained from five interactive methods for multi-objective optimization and three non-linear multi-objective test problems. These results suggest that the GR algorithm provides an efficient way to identify subsets of preferred Pareto optima from larger sets.

The main reason for applying evolutionary algorithms in multi-objective optimization problems is to obtain near-optimal nondominated solutions/Pareto fronts, from which decision-makers can choose a suitable solution. The efficiency of multi-objective optimization algorithms depends on the quality and quantity of Pareto fronts produced by them. To compare different Pareto fronts resulting from different algorithms, criteria are considered and applied in multi-objective problems. Each criterion denotes a characteristic of the Pareto front. Thus, ranking approaches are commonly used to evaluate different algorithms based on different criteria. This paper presents three multi-objective optimization methods based on the multi-objective particle swarm optimization (MOPSO) algorithm. To evaluate these methods, bi-objective mathematical benchmark problems are considered. Results show that all proposed methods are successful in finding near-optimal Pareto fronts. A ranking method is used to compare the capability of the proposed methods and the best method for further study is suggested. Moreover, the nominated method is applied as an optimization tool in real multi-objective optimization problems in multireservoir system operations. A new technique in multi-objective optimization, called warm-up, based on the PSO algorithm is then applied to improve the quality of the Pareto front by single-objective search. Results show that the proposed technique is successful in finding an optimal Pareto front.

A single front genetic algorithm for parallel multi-objective optimization in dynamic environments

Stochastic approach versus multiobjective approach for obtaining efficient solutions in stochastic multiobjective programming problems

: So far the study of stochastic programs with recourse has been limited to the case (called by G. Dantzig programming under uncertainty) when only the right-hand sides or resources of the problem are random. In this paper the authors extend the theory to the general case when essentially all the parameters involved are random. This generalization immediately raises the problem of attributing a precise meaning to the stochastic constraints. They examine a probability formulation (satisfying the constraints almost surely) and a possibility formulation (satisfying the constraints for all values of the random parameters in the support of their joint distribution) and show them equivalent under a rather weak but curious W-condition. Finally, they prove that without restriction the equivalent deterministic form of a stochastic program with recourse is a convex program for which we obtain some additional properties when some of the parameters of the original problem are constant. The applications of the theoretical results of this paper to certain classes of stochastic programs which have arisen from practical problems will be presented in a separate paper: 'Stochastic Programs with Recourse: Special Forms.' (Author)

SMGDA: an uncertainty based multi-objective optimization approach. Illustration to an airplane composite material

Many real-world design problems involve simultaneous optimisation of multiple objectives. The considered grillage design is a multi-objective optimisation problem since there are two objective functions, namely: the volume design and the cost design. Thus, the solution of this optimisation problem requires specialised method suitable for multi-objective problems. In this article, a multi-objective design optimisation using real-coded genetic algorithm is proposed for the Pareto-optimality of grillage system. The objective functions in the optimisation problem measure the design sensitivities in grillage system. The non-linear constrained multi-objective optimisation is very important from the point of view of practical problem solving. Therefore, the real-coded genetic algorithm with multiple genetic operators is proposed to find the optimum grillage system without handling any of the penalty functions. Direct strength calculation defined from the class rules of DNV was applied for structural design of grillage system. The hybrid method (real-coded genetic algorithm including the non-dominated sorting and sharing approaches) performs a marvellous explorability in finding a diverse set of solutions and in converging near the true Pareto-optimal set. The results obtained are very encouraging since they show that we can produce an important portion of the Pareto-front at a very low computational time frame.

Bayesian optimization is a versatile numerical method to solve global optimization problems of high complexity at a reduced computational cost. The efficiency of Bayesian optimization relies on two key elements: a surrogate model and an acquisition function. The surrogate model is generated on a Gaussian process statistical framework and provides probabilistic information of the prediction. The acquisition function, which guides the optimization, uses the surrogate probabilistic information to balance the exploration and the exploitation of the design space. In the case of multi-objective problems, current implementations use acquisition functions such as the multi-objective expected improvement (MEI). The evaluation of MEI requires a surrogate model for each objective function. In order to expand the Pareto front, such implementations perform a multi-variate integral over an intricate hypervolume, which require high computational cost. The objective of this work is to introduce an efficient multi-objective Bayesian optimization method that avoids the need for multi-variate integration. The proposed approach employs the working principle of multi-objective traditional methods, e.g., weighted sum and min-max methods, which transform the multi-objective problem into a single-objective problem through a functional mapping of the objective functions. Since only one surrogate is trained, this approach has a low computational cost. The effectiveness of the proposed approach is demonstrated with the solution of four problems: (1) an unconstrained version of the Binh and Korn test problem (convex Pareto front), (2) the Fonseca and Fleming test problem (non-convex Pareto front), (3) a three-objective test problem and (4) the design optimization of a sandwich composite armor for blast mitigation. The optimization algorithm is implemented in MATLAB and the finite element simulations are performed in the explicit, nonlinear finite element analysis code LS-DYNA. The results are comparable (or superior) to the results of the MEI acquisition function.

A multi-objective particle swarm optimization (MOPSO) approach for multi-objective economic load dispatch problem in power system is presented in this paper. The economic load dispatch problem is a non-linear constrained multi-objective optimization problem. The proposed MOPSO approach handles the problem as a multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives and has a diversity-preserving mechanism using an external memory (call “repository”) and a geographically-based approach to find widely different Pareto-optimal solutions. In addition, fuzzy set theory is employed to extract the best compromise solution. Several optimization runs of the proposed MOPSO approach were carried out on the standard IEEE 30-bus test system. The results revealed the capabilities of the proposed MOPSO approach to generate well-distributed Pareto-optimal non-dominated solutions of multi-objective economic load dispatch. Comparison with Multi-objective Evolutionary Algorithm (MOEA) showed the superiority of the proposed MOPSO approach and confirmed its potential for solving multi-objective economic load dispatch.

The optimization problems are taking place at all times in actual lives. They are divided into single objective problems and multiobjective problems. Single objective optimization has only one objective function, while multiobjective optimization has multiple objective functions that generate the Pareto set. Therefore, to solve multiobjective problems is a challenging task. A multiobjective particle swarm optimization, which combined cosine distance measurement mechanism and novel game strategy, has been proposed in this article. The cosine distance measurement mechanism was adopted to update Pareto optimal set in the external archive. At the same time, the candidate set was established so that Pareto optimal set deleted from the external archive could be effectively replaced, which helped to maintain the size of the external archive and improved the convergence and diversity of the swarm. In order to strengthen the selection pressure of leader, this article combined with the game update mechanism, and a global leader selection strategy that integrates the game strategy including the cosine distance mechanism was proposed. In addition, mutation was used to maintain the diversity of the swarm and prevent the swarm from prematurely converging to the true Pareto front. The performance of the proposed competitive multiobjective particle swarm optimizer was verified by benchmark comparisons with several state-of-the-art multiobjective optimizer, including seven multiobjective particle swarm optimization algorithms and seven multiobjective evolutionary algorithms. Experimental results demonstrate the promising performance of the proposed algorithm in terms of optimization quality.

In this paper we propose a new metaheuristic algorithm for solving stochastic multiobjective combinatorial optimization(SMOCO) problems. Indeed, we find that the various initiatives that have been launched recently on this subject, they propose the classical metaheuristics to solve a stochastic multi-objective problems,but when the stochasticity effects is not taken into account, the choice of an arbitrary value leading to a particular configuration and a high loss of information. To conserve the stochastic nature of SMOCO problems,the pareto ranking should be defined on the random objectives functions directly rather than converted deterministic objectives. From these considerations, the scope of this research should consist of: (i)Proposed novel methodology of stochastic optimality for ranking objective functions characterized by non-continuous and no closed form expression. This novel approach is based on combinatorial probability and can be incorporated in a multiobjective evolutionary algorithm. (ii)Provide probabilistic approaches to elitism and diversification in multiobjective evolutionary algorithms. Finally, The behavior of the resulting Probabilistic Multi-objective Evolutionary Algorithms (PrMOEAs) is empirically investigated on the multi-objective stochastic VRP problem.

Real-world optimization problems typically include uncertainties over various aspects of the problem formulation. Some existing algorithms are designed to cope with stochastic multiobjective optimization problems, but in order to benchmark them, a proper framework still needs to be established. This paper presents a novel toolkit that generates scalable, stochastic, multiobjective optimization problems. A stochastic problem is generated by transforming the objective vectors of a given deterministic test problem into random vectors. All random objective vectors are bounded by the feasible objective space, defined by the deterministic problem. Therefore, the global solution for the deterministic problem can also serve as a reference for the stochastic problem. A simple parametric distribution for the random objective vector is defined in a radial coordinate system, allowing for direct control over the dual challenges of convergence towards the true Pareto front and diversity across the front. An example for a stochastic test problem, generated by the toolkit, is provided.

The challenge of solving dynamic multi-objective optimization problems is to trace the varying Pareto optimal front and/or Pareto optimal set quickly and efficiently. This paper proposes a multi-direction prediction strategy using a hybrid chemical reaction optimization, aimed at finding the dynamic Pareto optimal front and/or Pareto optimal set as quickly and accurately as possible before the next environmental change occurs. The proposed method, multi-direction prediction multi-objective hybrid chemical reaction optimization algorithm which mainly includes a hybrid chemical reaction optimization algorithm is proposed for solving dynamic multi-objective problems, which can guide population trace the optimum. When the environment has changed, the population is divided into several subpopulations. In subpopulation, a center point was found to construct multi-direction prediction model. As a result, this approach enhances the diversity of algorithm. While the environment has not changed, the hybrid chemical reaction algorithm with particle swarm optimization algorithm can efficiency find the optimal solution, it can achieve good diversity as well as guarantee the avoidance of local optimal solutions. The proposed algorithm is measured on several benchmark test suites with various dynamic characteristics and different difficulties. Experimental results show that this algorithm is very competitive in dealing with dynamic multi-objective optimization problems when compared with four state-of-the-art approaches.KeywordsDynamic multi-objective optimizationHybrid chemical reaction optimization algorithmMulti-direction prediction