Structure and Weak Sharp Minimum of the Pareto Solution Set for Piecewise Linear Multiobjective Optimization

In this paper, the Pareto solution set of a piecewise linear multiobjective optimization problem in a normed space is shown to be the union of finitely many semiclosed polyhedra. If the problem is further assumed to be cone-convex, then it has the global weak sharp minimum property.     

The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces

In general normed spaces,we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior.We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function.Moreover,we provide necessary and suffcient conditions about the existence of weak(sharp) Pareto solutions.     

Convergence analysis of Tikhonov-type regularization algorithms for multiobjective optimization problems

In this paper, we introduce a vector-valued Tikhonov-type regularization algorithm for an extended-valued multiobjective optimization problem. Under some mild conditions, we prove that any sequence generated by this algorithm converges to a weak Pareto optimal solution of the multiobjective optimization problem. Our results improve and generalize some known results.     

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.     

Global weak sharp minima for convex (semi-)infinite optimization problems

We mainly consider global weak sharp minima for convex infinite and semi-infinite optimization problems (CIP). In terms of the normal cone, subdifferential and directional derivative, we provide several characterizations for (CIP) to have global weak sharp minimum property.     

UNIFORM APPROXIMATION OF A MULTIOBJECTIVE OPTIMIZATION SEQUENCE

In this paper the Pareto efficiency of a uniformly convergent multiobjective optimization sequence is studied. We obtain some relation between the Pareto efficient solutions of a given multiobjective optimization problem and those of its uniformly convergent optimization sequence and also some relation between the weak Pareto efficient solutions of the same optimization problem and those of its uniformly convergent optimization sequence. Besides, under a compact convex assumption for constraints set and a certain convex assumption for both objective and constraint functions, we also get some sufficient and necessary conditions that the limit of solutions of a uniformly convergent multiobjective optimization sequence is the solution of a given multiobjective optimization problem.     

Generalized viscosity approximation methods in multiobjective optimization problems

In this paper, we consider an extend-valued nonsmooth multiobjective optimization problem of finding weak Pareto optimal solutions. We propose a class of vector-valued generalized viscosity approximation method for solving the problem. Under some conditions, we prove that any sequence generated by this method converges to a weak Pareto optimal solution of the multiobjective optimization problem.     

Weak sharp efficiency in multiobjective optimization

By using the generalized Fermat rule, the Mordukhovich subdifferential for maximum functions, the fuzzy sum rule for Fréchet subdifferentials and the sum rule for Mordukhovich subdifferentials, we establish a necessary optimality condition for the local weak sharp efficient solution of a constrained multiobjective optimization problem. Moreover, by employing the approximate projection theorem, and some appropriate convexity and affineness conditions, we also obtain some sufficient optimality conditions respectively for the local and global weak sharp efficient solutions of such a multiobjective optimization problem.     

On a Sufficient Condition for Weak Sharp Efficiency in Multiobjective Optimization

In this paper, we provide sufficient conditions entailing the existence of weak sharp efficient points of a multiobjective optimization problem. The approach uses variational analysis techniques, like regularity and subregularity of the diagonal subdifferential map related to a suitable scalar equilibrium problem naturally associated to the multiobjective optimization problem.     

Generalized asymptotic functions in nonconvex multiobjective optimization problems

In this paper, we use generalized asymptotic functions and second-order asymptotic cones to develop a general existence result for the nonemptiness of the proper efficient solution set and a sufficient condition for the domination property in nonconvex multiobjective optimization problems. A new necessary condition for a point to be efficient or weakly efficient solution is given without any convexity assumption. We also provide a finer outer estimate for the asymptotic cone of the weakly efficient solution set in the quasiconvex case. Finally, we apply our results to the linear fractional multiobjective optimization problem.     

非光滑非凸多目标规划解的充分条件

Kuhn-Tucker型条件的充分性一直是最优化理论中引人注意的一个问题.本文对非光滑函数提出了几个非凸概念,然后,讨论了非光滑非凸多目标规划中Kuhn-Tucker型条件和Fritz John型条件的充分性,在很弱的条件下,建立了一系列充分条件.     

A portfolio theory approach to crop planning under environmental constraints

This paper presents a multiobjective model for crop planning in agriculture. The approach is based on portfolio theory. The model takes into account weather risks, market risks and environmental risks. Input data include historical land productivity data for various crops, soil types and yield response to fertilizer/pesticide application. Several environmental levels for the application of fertilizers/pesticides, and the monetary penalties for overcoming these levels, are also considered. Starting from the multiobjective model we formulate several single objective optimization problems: the minimum environmental risk problem, the maximum expected return problem and the minimum financial risk problem. We prove that the minimum environmental risk problem is equivalent to a mixed integer problem with a linear objective function. Two numerical results for the minimum environmental risk problem are presented.     

Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets

In this paper, we investigate the separations and optimality conditions for the optimal solution defined by the improvement set of a constrained multiobjective optimization problem. We introduce a vector-valued regular weak separation function and a scalar weak separation function via a nonlinear scalarization function defined in terms of an improvement set. The nonlinear separation between the image of the multiobjective optimization problem and an improvement set in the image space is established by the scalar weak separation function. Saddle point type optimality conditions for the optimal solution of the multiobjective optimization problem are established, respectively, by the nonlinear and linear separation methods. We also obtain the relationships between the optimal solution and approximate efficient solution of the multiobjective optimization problem. Finally, sufficient and necessary conditions for the (regular) linear separation between the approximate image of the multiobjective optimization problem and a convex cone are also presented.     

Weak Subdifferential of Set-Valued Mappings

In this note we introduce a notion of the weak contingent generalized gradient for set-valued mappings associated with the contingent epiderivative of set-valued mappings introduced in "E. Bednarczuk and W. Song (1998). Contingent epiderivative and its applications to set-valued optimization. Control and Cybernetics, 27, 376-386; G.Y. Chen and J. Jahn (1998). Optimally conditions for set-valued optimization problems. Mathematical Methods of Operations Research, 48, 187-200." and prove that, under some additional condition, it coincides with the weak subdifferential introduced in "T. Tanino (1992). Conjugate duality in vector optimization. Journal of Mathematical Analysis and Applications, 167, 84-97." when the set-valued map is cone-convex. We also study the weak contingent generalized gradient of a sum of two set-valued mappings and optimality conditions for a set-valued vector optimization problem.     

On Weak and Strong Kuhn–Tucker Conditions for Smooth Multiobjective Optimization

We consider a smooth multiobjective optimization problem with inequality constraints. Weak Kuhn?CTucker (WKT) optimality conditions are said to hold for such problems when not all the multipliers of the objective functions are zero, while strong Kuhn?CTucker (SKT) conditions are said to hold when all the multipliers of the objective functions are positive. We introduce a new regularity condition under which (WKT) hold. Moreover, we prove that for another new regularity condition (SKT) hold at every Geoffrion-properly efficient point. We show with an example that the assumption on proper efficiency cannot be relaxed. Finally, we prove that Geoffrion-proper efficiency is not needed when the constraint set is polyhedral and the objective functions are linear.     

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.     

Existence of solutions for vector optimization

In this paper, we prove the existence of a weak minimum for constrained vector optimization problem by making use of vector variational-like inequality and preinvex functions.     

Stability of Local Efficiency in Multiobjective Optimization

Analyzing the behavior and stability properties of a local optimum in an optimization problem, when small perturbations are added to the objective functions, are important considerations in optimization. The tilt stability of a local minimum in a scalar optimization problem is a well-studied concept in optimization which is a version of the Lipschitzian stability condition for a local minimum. In this paper, we define a new concept of stability pertinent to the study of multiobjective optimization problems. We prove that our new concept of stability is equivalent to tilt stability when scalar optimizations are available. We then use our new notions of stability to establish new necessary and sufficient conditions on when strict locally efficient solutions of a multiobjective optimization problem will have small changes when correspondingly small perturbations are added to the objective functions.     

Inner and outer estimates for the solution sets and their asymptotic cones in vector optimization

We use asymptotic analysis to develop finer estimates for the efficient, weak efficient and proper efficient solution sets (and for their asymptotic cones) to convex/quasiconvex vector optimization problems. We also provide a new representation for the efficient solution set without any convexity assumption, and the estimates involve the minima of the linear scalarization of the original vector problem. Some new necessary conditions for a point to be efficient or weak efficient solution for general convex vector optimization problems, as well as for the nonconvex quadratic multiobjective optimization problem, are established.     

A path following method for box-constrained multiobjective optimization with applications to goal programming problems

We propose a path following method to find the Pareto optimal solutions of a box-constrained multiobjective optimization problem. Under the assumption that the objective functions are Lipschitz continuously differentiable we prove some necessary conditions for Pareto optimal points and we give a necessary condition for the existence of a feasible point that minimizes all given objective functions at once. We develop a method that looks for the Pareto optimal points as limit points of the trajectories solutions of suitable initial value problems for a system of ordinary differential equations. These trajectories belong to the feasible region and their computation is well suited for a parallel implementation. Moreover the method does not use any scalarization of the multiobjective optimization problem and does not require any ordering information for the components of the vector objective function. We show a numerical experience on some test problems and we apply the method to solve a goal programming problem.