Multiobjective DC programs with infinite convex constraints

New results are established for multiobjective DC programs with infinite convex constraints (MOPIC) that are defined on Banach spaces (finite or infinite dimensional) with objectives given as the difference of convex functions. This class of problems can also be called multiobjective DC semi-infinite and infinite programs, where decision variables run over finite-dimensional and infinite-dimensional spaces, respectively. Such problems have not been studied as yet. Necessary and sufficient optimality conditions for the weak Pareto efficiency are introduced. Further, we seek a connection between multiobjective linear infinite programs and MOPIC. Both Wolfe and Mond-Weir dual problems are presented, and corresponding weak, strong, and strict converse duality theorems are derived for these two problems respectively. We also extend above results to multiobjective fractional DC programs with infinite convex constraints. The results obtained are new in both semi-infinite and infinite frameworks.     

Nonsmooth multiobjective programming with quasi-Newton methods

This paper proposes a new algorithm to solve nonsmooth multiobjective programming. The algorithm is a descent direction method to obtain the critical point (a necessary condition for Pareto optimality). We analyze both global and local convergence results under some assumptions. Numerical tests are also given.     

Multiple reduced gradient method for multiobjective optimization problems

Multiobjective optimization has a large number of real-life applications. Under this motivation, in this paper, we present a new method for solving multiobjective optimization problems with both linear constraints and bound constraints on the variables. This method extends, to the multiobjective setting, the classical reduced gradient method for scalar-valued optimization. The proposed algorithm generates a feasible descent direction by solving an appropriate quadratic subproblem, without the use of any scalarization approaches. We prove that the sequence generated by the algorithm converges to Pareto-critical points of the problem. We also present some numerical results to show the efficiency of the proposed method.     

The influence of the fitness evaluation method on the performance of multiobjective search algorithms

In this paper we are concerned with finding the Pareto optimal front or a good approximation to it. Since non-dominated solutions represent the goal in multiobjective optimisation, the dominance relation is frequently used to establish preference between solutions during the search. Recently, relaxed forms of the dominance relation have been proposed in the literature for improving the performance of multiobjective search methods. This paper investigates the influence of different fitness evaluation methods on the performance of two multiobjective methodologies when applied to a highly constrained two-objective optimisation problem. The two algorithms are: the Pareto archive evolutionary strategy and a population-based annealing algorithm. We demonstrate here, on a highly constrained problem, that the method used to evaluate the fitness of candidate solutions during the search affects the performance of both algorithms and it appears that the dominance relation is not always the best method to use.     

A new reduced gradient method for solving linearly constrained multiobjective optimization problems

In this paper, we consider the linearly constrained multiobjective minimization, and we propose a new reduced gradient method for solving this problem. Our approach solves iteratively a convex quadratic optimization subproblem to calculate a suitable descent direction for all the objective functions, and then use a bisection algorithm to find an optimal stepsize along this direction. We prove, under natural assumptions, that the proposed algorithm is well-defined and converges globally to Pareto critical points of the problem. Finally, this algorithm is implemented in the MATLAB environment and comparative results of numerical experiments are reported.     

A weighting subgradient algorithm for multiobjective optimization

We propose a weighting subgradient algorithm for solving multiobjective minimization problems on a nonempty closed convex subset of an Euclidean space. This method combines weighting technique and the classical projected subgradient method, using a divergent series steplength rule. Under the assumption of convexity, we show that the sequence generated by this method converges to a Pareto optimal point of the problem. Some numerical results are presented.     

A comparison of feasible direction methods for the stochastic transportation problem

The feasible direction method of Frank and Wolfe has been claimed to be efficient for solving the stochastic transportation problem. While this is true for very moderate accuracy requirements, substantially more efficient algorithms are otherwise diagonalized Newton and conjugate Frank–Wolfe algorithms, which we describe and evaluate. Like the Frank–Wolfe algorithm, these two algorithms take advantage of the structure of the stochastic transportation problem. We also introduce a Frank–Wolfe type algorithm with multi-dimensional search; this search procedure exploits the Cartesian product structure of the problem. Numerical results for two classic test problem sets are given. The three new methods that are considered are shown to be superior to the Frank–Wolfe method, and also to an earlier suggested heuristic acceleration of the Frank–Wolfe method.     

A new subspace minimization conjugate gradient method with nonmonotone line search for unconstrained optimization

A new subspace minimization conjugate gradient algorithm with a nonmonotone Wolfe line search is proposed and analyzed. In the scheme, we propose two choices of the search direction by minimizing a quadratic approximation of the objective function in special subspaces, and state criterions on how to choose the direction. Under given conditions, we obtain the significant conclusion that each choice of the direction satisfies the sufficient descent property. Based on the idea on how the function is close to a quadratic function, a new strategy for choosing the initial stepsize is presented for the line search. With the used nonmonotone Wolfe line search, we prove the global convergence of the proposed method for general nonlinear functions under mild assumptions. Numerical comparisons are given with well-known CGOPT and CG_DESCENT and show that the proposed algorithm is very promising.     

Multiobjective shortest path problems with lexicographic goal-based preferences

Multiobjective shortest path problems are computationally harder than single objective ones. In particular, execution time is an important limiting factor in exact multiobjective search algorithms. This paper explores the possibility of improving search performance in those cases where the interesting portion of the Pareto front can be initially bounded. We introduce a new exact label-setting algorithm that returns the subset of Pareto optimal paths that satisfy a set of lexicographic goals, or the subset that minimizes deviation from goals if these cannot be fully satisfied. Formal proofs on the correctness of the algorithm are provided. We also show that the algorithm always explores a subset of the labels explored by a full Pareto search. The algorithm is evaluated over a set of problems with three objectives, showing a performance improvement of up to several orders of magnitude as goals become more restrictive.     

Another Conjugate Gradient Algorithm with Guaranteed Descent and Conjugacy Conditions for Large-scale Unconstrained Optimization

In this paper, we suggest another accelerated conjugate gradient algorithm for which both the descent and the conjugacy conditions are guaranteed. The search direction is selected as a linear combination of the gradient and the previous direction. The coefficients in this linear combination are selected in such a way that both the descent and the conjugacy condition are satisfied at every iteration. The algorithm introduces the modified Wolfe line search, in which the parameter in the second Wolfe condition is modified at every iteration. It is shown that both for uniformly convex functions and for general nonlinear functions, the algorithm with strong Wolfe line search generates directions bounded away from infinity. The algorithm uses an acceleration scheme modifying the step length in such a manner as to improve the reduction of the function values along the iterations. Numerical comparisons with some conjugate gradient algorithms using a set of 75 unconstrained optimization problems with different dimensions show that the computational scheme outperforms the known conjugate gradient algorithms like Hestenes and Stiefel; Polak, Ribière and Polyak; Dai and Yuan or the hybrid Dai and Yuan; CG_DESCENT with Wolfe line search, as well as the quasi-Newton L-BFGS.     

MultiGLODS: global and local multiobjective optimization using direct search

The optimization of multimodal functions is a challenging task, in particular when derivatives are not available for use. Recently, in a directional direct search framework, a clever multistart strategy was proposed for global derivative-free optimization of single objective functions. The goal of the current work is to generalize this approach to the computation of global Pareto fronts for multiobjective multimodal derivative-free optimization problems. The proposed algorithm alternates between initializing new searches, using a multistart strategy, and exploring promising subregions, resorting to directional direct search. Components of the objective function are not aggregated and new points are accepted using the concept of Pareto dominance. The initialized searches are not all conducted until the end, merging when they start to be close to each other. The convergence of the method is analyzed under the common assumptions of directional direct search. Numerical experiments show its ability to generate approximations to the different Pareto fronts of a given problem.     

Combination of steepest descent and BFGS methods for nonconvex nonsmooth optimization

In this paper, a method is developed for solving nonsmooth nonconvex minimization problems. This method extends the classical BFGS framework. First, we generalize the Wolfe conditions for locally Lipschitz functions and prove that this generalization is well defined. Then, a line search algorithm is presented to find a step length satisfying the generalized Wolfe conditions. Next, the Goldstein e-subgradient is approximated by an iterative method and a descent direction is computed using a positive definite matrix. This matrix is updated using the BFGS method. Finally, a minimization algorithm based on the BFGS method is described. The algorithm is implemented in MATLAB and numerical results using it are reported.     

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.     

NEW HYBRID CONJUGATE GRADIENT METHOD AS A CONVEX COMBINATION OF LS AND FR METHODS

In this paper, we present a new hybrid conjugate gradient algorithm for unconstrained optimization. This method is a convex combination of Liu-Storey conjugate gradient method and Fletcher-Reeves conjugate gradient method. We also prove that the search direction of any hybrid conjugate gradient method, which is a convex combination of two conjugate gradient methods, satisfies the famous D-L conjugacy condition and in the same time accords with the Newton direction with the suitable condition. Furthermore, this property doesn't depend on any line search. Next, we also prove that, moduling the value of the parameter t,the Newton direction condition is equivalent to Dai-Liao conjugacy condition.The strong Wolfe line search conditions are used.The global convergence of this new method is proved.Numerical comparisons show that the present hybrid conjugate gradient algorithm is the efficient one.     

Variable and large neighborhood search to solve the multiobjective set covering problem

In this paper, we study the multiobjective version of the set covering problem. To our knowledge, this problem has only been addressed in two papers before, and with two objectives and heuristic methods. We propose a new heuristic, based on the two-phase Pareto local search, with the aim of generating a good approximation of the Pareto efficient solutions. In the first phase of this method, the supported efficient solutions or a good approximation of these solutions is generated. Then, a neighborhood embedded in the Pareto local search is applied to generate non-supported efficient solutions. In order to get high quality results, two elaborate local search techniques are considered: a large neighborhood search and a variable neighborhood search. We intensively study the parameters of these two techniques. We compare our results with state-of-the-art results and we show that with our method, better results are obtained for different indicators.     

A method for generating a well-distributed Pareto set in nonlinear multiobjective optimization

A method is presented for generating a well-distributed Pareto set in nonlinear multiobjective optimization. The approach shares conceptual similarity with the Physical Programming-based method, the Normal-Boundary Intersection and the Normal Constraint methods, in its systematic approach investigating the objective space in order to obtain a well-distributed Pareto set. The proposed approach is based on the generalization of the class functions which allows the orientation of the search domain to be conducted in the objective space. It is shown that the proposed modification allows the method to generate an even representation of the entire Pareto surface. The generation is performed for both convex and nonconvex Pareto frontiers. A simple algorithm has been proposed to remove local Pareto solutions. The suggested approach has been verified by several test cases, including the generation of both convex and concave Pareto frontiers.     

A Proximal Point-Type Method for Multicriteria Optimization

In this paper, we present a proximal point algorithm for multicriteria optimization, by assuming an iterative process which uses a variable scalarization function. With respect to the convergence analysis, firstly we show that, for any sequence generated from our algorithm, each accumulation point is a Pareto critical point for the multiobjective function. A more significant novelty here is that our paper gets full convergence for quasi-convex functions. In the convex or pseudo-convex cases, we prove convergence to a weak Pareto optimal point. Another contribution is to consider a variant of our algorithm, obtaining the iterative step through an unconstrained subproblem. Then, we show that any sequence generated by this new algorithm attains a Pareto optimal point after a finite number of iterations under the assumption that the weak Pareto optimal set is weak sharp for the multiobjective problem.     

Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization

The use of surrogate based optimization (SBO) is widely spread in engineering design to reduce the number of computational expensive simulations. However, “real-world” problems often consist of multiple, conflicting objectives leading to a set of competitive solutions (the Pareto front). The objectives are often aggregated into a single cost function to reduce the computational cost, though a better approach is to use multiobjective optimization methods to directly identify a set of Pareto-optimal solutions, which can be used by the designer to make more efficient design decisions (instead of weighting and aggregating the costs upfront). Most of the work in multiobjective optimization is focused on multiobjective evolutionary algorithms (MOEAs). While MOEAs are well-suited to handle large, intractable design spaces, they typically require thousands of expensive simulations, which is prohibitively expensive for the problems under study. Therefore, the use of surrogate models in multiobjective optimization, denoted as multiobjective surrogate-based optimization, may prove to be even more worthwhile than SBO methods to expedite the optimization of computational expensive systems. In this paper, the authors propose the efficient multiobjective optimization (EMO) algorithm which uses Kriging models and multiobjective versions of the probability of improvement and expected improvement criteria to identify the Pareto front with a minimal number of expensive simulations. The EMO algorithm is applied on multiple standard benchmark problems and compared against the well-known NSGA-II, SPEA2 and SMS-EMOA multiobjective optimization methods.     

Multiple subgradient descent bundle method for convex nonsmooth multiobjective optimization

The aim of this paper is to propose a new multiple subgradient descent bundle method for solving unconstrained convex nonsmooth multiobjective optimization problems. Contrary to many existing multiobjective optimization methods, our method treats the objective functions as they are without employing a scalarization in a classical sense. The main idea of this method is to find descent directions for every objective function separately by utilizing the proximal bundle approach, and then trying to form a common descent direction for every objective function. In addition, we prove that the method is convergent and it finds weakly Pareto optimal solutions. Finally, some numerical experiments are considered.     

Multicriteria optimization with a multiobjective golden section line search

This work presents an algorithm for multiobjective optimization that is structured as: (i) a descent direction is calculated, within the cone of descent and feasible directions, and (ii) a multiobjective line search is conducted over such direction, with a new multiobjective golden section segment partitioning scheme that directly finds line-constrained efficient points that dominate the current one. This multiobjective line search procedure exploits the structure of the line-constrained efficient set, presenting a faster compression rate of the search segment than single-objective golden section line search. The proposed multiobjective optimization algorithm converges to points that satisfy the Kuhn-Tucker first-order necessary conditions for efficiency (the Pareto-critical points). Numerical results on two antenna design problems support the conclusion that the proposed method can solve robustly difficult nonlinear multiobjective problems defined in terms of computationally expensive black-box objective functions.