Approximate solutions of vector optimization problems

Existence of Solutions for Vector Optimization Problems

Generalized properly efficient solutions of a vector optimization problem (VP) are defined in terms of various tangent cones and a generalized directional derivative. We study their basic properties and relationships and show that under certain conditions, a generalized properly efficient solution of (VP), defined by the adjacent cone, is a generalized Kuhn-Tucker properly efficient solution of (VP). Furthermore, using subgradients defined by closed convex tangent cones, we give a necessary optimality condition for a generalized properly efficient solution of (VP) defined by the adjacent cone.

In this paper, we establish several proper separation theorems for an element and a convex set and for two convex sets in terms of their quasi-relative interiors. Then, we prove that the separation theorem given by [Cammaroto, F and B Di Bella (2007). On a separation theorem involving the quasi-relative. Proceedings of the Edinburgh Mathematical Society, 50(3), 605–610] in Theorem 2.5, is in fact a proper separation theorem for two convex sets in which the classical interior is replaced by the quasi-relative interior. Besides, we extend some known results in the literature, such as [Adán, M and V Novo (2004). Proper efficiency in vector optimization on real linear spaces. Journal of Optimization Theory and Applications, 121, 515–540] in Theorem 2.1 and [Edwards, R (1965). Functional Analysis: Theory and Applications. New York: Reinhart and Winston] in Corollary 2.2.2, through the quasi-relative interior and the quasi-interior, respectively. As an application, we provide Karush–Kuhn–Tucker multipliers for quasi-relative solutions of vector optimization problems. Several examples are given to illustrate the obtained results.

In this paper, we introduce a new kind of properly approximate efficient solution of vector optimization problems. Some properties for this new classof approximate solutions are established. Also necessary and sufficient conditions via nonlinear scalarizations areobtained for properly approximate solutions. And under theassumption of cone subconvexlike functions, we derive linearscalarizations for properly approximate efficient solutions.

The aim of this article is to provide necessary and sufficient conditions for a feasible point to be a weakly ϵ-efficient solution of a vector optimization problem with cone constraints under generalized convexity notions and a weak constraint qualification. Our results permit to recover earlier established theorems.

In this paper, K- quasiconvex, K- pseudoconvex and other related functions have been introduced in terms of their Clarke subdifferentials, where is an arbitrary closed convex, pointed cone with nonempty interior. The (strict, weakly) -pseudomonotonicity, (strict) K- naturally quasimonotonicity and K- quasimonotonicity of Clarke subdifferential maps have also been defined. Further, we introduce Minty weak (MVVIP) and Stampacchia weak (SVVIP) vector variational inequalities over arbitrary cones. Under regularity assumption, we have proved that a weak minimum solution of vector optimization problem (VOP) is a solution of (SVVIP) and under the condition of K- pseudoconvexity we have obtained the converse for MVVIP (SVVIP). In the end we study the interrelations between these with the help of strict K-naturally quasimonotonicity of Clarke subdifferential map.

The aim of this paper is to study the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving (G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of (G, α)-invex functions. Examples are provided to elucidate our results.

In this paper, I introduce two kinds of ε-subgradients—ε-Pareto subgradients and ε-Benson proper subgradients—for set-valued maps. I give sufficient conditions for the existence of such ε-subgradients. A generalized ε-Moreau–Rockafellar type theorem for ε-Benson proper subdifferentials of set-valued maps is formulated and applied to establish ε-optimality conditions for ε-Benson proper efficient solutions of vector optimization problems with set-valued maps. Scalarization theorems for such a problem are also obtained.

Generalized gradient projection neural network models are proposed to solve nonsmooth convex and nonconvex nonlinear programming problems over a closed convex subset of ℝ n . By using Clarke’s generalized gradient, the neural network modeled by a differential inclusion is developed, and its dynamical behavior and optimization capabilities both for convex and nonconvex problems are rigorously analyzed in the framework of nonsmooth analysis and the differential inclusion theory. First for nonconvex optimization problems, the quasiconvergence results similar to those achieved by previous work are proved. For convex optimization problems, the global convergence results are proved, i.e. any trajectory of the neural network converges to an asymptotically stable equilibrium point, which is an optimal solution of the primal problem whenever it exists. This result remains valid even if the initial condition is chosen outside the feasible set. In addition, an asymptotic control result involving a Tykhonov-like regularization shows that any trajectory of the revised neural network can be forced to converge toward a particular optimal solution of the primal problem. Finally, two simulation optimization algorithms for solving the optimization and control problems are designed respectively, and three typical simulation experiments are illustrated to present the accuracy and efficiency of theoretical convergence results of this paper.

A numerical multi-level optimization methodology is proposed for determining dextrous workspaces of 3-degree-of-freedom (3-dof) planar parallel manipulators, in which it is required that at any point within the workspace, the manipulator is able to assume any orientation in a specified range. The method starts by finding a single initial point on the boundary of the dextrous workspace. This first stage requires the successive solution of three separate optimization sub-problems, where the evaluation of the objective function for the second problem and the constraint functions in the third problem are determined by the solution of appropriate optimization problems at a lower level. Once the boundary point is identified, further successive points along the dextrous workspace boundary are traced by the application of the so-called chord method. In the latter procedure, the determination of each successive boundary point is also obtained via a constrained optimization problem, where the constraint functions are again evaluated via the solution of an optimization problem at a lower level. The proposed method is illustrated by its successful application to three different manipulator design geometries, and for various ranges of dexterity.

We take into condideration necessary optimality conditions of minimum principle-type, that is for optimization problems having, besides the usual inequality and/or equality constraints, a set constraint. The first part pf the paper is concerned with scalar optimization problems; the second part of the paper deals with vector optimization problems.

This paper discusses the optimality conditions for the strong efficient solutions of set-valued optimization problems. Firstly, by using the second-order weak subdifferential of set-valued mappings and the relationship between the strong efficient solutions of set-valued optimization problems and the optimal solutions of composite optimization problems, one establishes the optimality necessary and sufficient conditions of the strong efficient solutions for unconstrained set-valued optimization problems. Meanwhile, by using the second-order weak subdifferential of set- valued mappings, several optimality necessary and sufficient conditions of unconstrained composite optimization problems are obtained under the assumption of generalized cone sub-convexlikeness. Secondly, one introduces a class of saddle points for constrained set-valued optimization problems. Then, several sufficient optimality conditions for strong efficient solutions of constrained set-valued optimization problems are established under the assumptions of generalized cone sub-convexlikeness and saddle points for constrained set-valued optimization problems. Additionally, by using the second-order weak subdifferential of set-valued mappings, the sufficient optimality conditions for strong efficient solutions of constrained set-valued optimization problems are established under the assumption of generalized cone sub-convexlikeness. Finally, two optimality sufficient conditions for the optimal solutions of the constrained composite optimization problem are obtained under the assumption of generalized cone sub-convexlikeness. 主要讨论集值优化问题强有效解的最优性条件。首先，借助于集值映射的二阶弱次微分，利用集合优化问题强有效解与复合优化问题最优解的关系，在广义锥次似凸假设条件下建立了无约束集值优化问题强有效解的最优性必要条件和充分条件。同时，利用集值映射的二阶弱次微分，在广义锥次似凸假设条件下也获得了无约束复合优化问题的几个最优性必要条件和充分条件。其次，引入了约束集值优化问题的一类鞍点，并在广义锥次似凸和约束集值优化问题鞍点的假设条件下，建立了约束集值优化问题强有效解的最优性充分条件。另外，借助于集值映射的二阶弱次微分，在广义锥次似凸假设条件下，建立了约束集值优化问题强有效解的最优性充分条件。最后，在广义锥次似凸假设条件下，获得了约束复合优化问题最优解的2个最优性充分条件。

We present an approach for efficient numerical solution of optimization problems governed by parabolic partial differential equations. The main ingredients are: space-time finite element discretization, second-order optimization algorithms, and storage reduction techniques. We discuss the combination of these components for the solution of large-scale optimization problems.

The aim of this article is to apply and improve the species-conserving genetic algorithm (SCGA) to search multiple solutions of truss topology optimization problems in a single run. A species is defined as a group of individuals with similar characteristics and is dominated by its species seed. The solutions of an optimization problem will be selected from the found species. To improve the accuracy of solutions, a species mutation technique is introduced to improve the fitness of the found species seeds and the combination of a neighbour mutation and a uniform mutation is applied to balance exploitation and exploration. A real vector is used to represent the corresponding cross-sectional areas and a member is thought to be existent if its area is bigger than a critical area. A finite element analysis model was developed to deal with more practical considerations in modelling, such as the existence of members, kinematic stability analysis, and computation of stresses and displacements. Cross-sectional areas and node connections are decision variables and optimized simultaneously to minimize the total weight of trusses. Numerical results demonstrate that some truss topology optimization examples have many global and local solutions, different topologies can be found using the proposed algorithm on a single run and some trusses have smaller weights than the solutions in the literature.

The numerical solution of optimization problems with partial differential equation (PDE) constraints is vital to a growing number of science and engineering applications. The development of robust and efficient algorithms for the solution of these optimization problems presents many challenges that arise out of, e.g., the intricate mathematical structure of these problems, the complicated interactions between numerical methods for PDE and optimization, the large-scale of the optimization problems, and the increasing complexity of applications. To identify and overcome these challenges an integrated approach is needed that builds on a variety of mathematical sub-disciplines, such as theory of PDEs, distributed parameter systems, numerical solution of PDEs, numerical optimization, and numerical linear algebra. This international workshop has brought together some of the leading experts in the fast developing field of optimization problems with PDE constraints to present recent developments in this area as well as to identify open problems and further research needs. Among the themes of this workshop were the design and analysis of approaches for the solution of PDE constrained optimization problems with additional point-wise constraints on controls and states (the solution of the governing PDE). State constrained problems are particularly challenging because of the low regularity properties of the Lagrange multipliers associated with point-wise constraints on the states. A second theme was the development of adaptive methods for the solution of PDE constrained optimization problems and, more generally, the development of optimization level model reduction techniques for these problems. The goal here is to develop models (through, e.g., mesh adaptation or proper orthogonal model reduction) of the PDE constrained optimization problems that capture the relevant features of the optimization problems with a specified accuracy, but involve as few degrees of freedom as possible and, hence, are computationally less expensive to work with. A third theme was The efficient solution of linear systems arising in optimization algorithms for discretized PDE constrained optimization problems represented another theme. Finally, a number of talks presented advances and challenges in the solution of PDE constrained optimization problems arising in important science and industrial applications.