Scalarization of Levitin–Polyak well-posed set optimization problems

The aim of this paper is to study Levitin–Polyak (LP in short) well-posedness for set optimization problems. We define the global notions of metrically well-setness and metrically LP well-setness and the pointwise notions of LP well-posedness, strongly DH-well-posedness and strongly B-well-posedness for set optimization problems. Using a scalarization function defined by means of the point-to-set distance, we characterize the LP well-posedness and the metrically well-setness of a set optimization problem through the LP well-posedness and the metrically well-setness of a scalar optimization problem, respectively.     

A matrix partition problem

In the set of positive definite semi-integral symmetric matrices we propose a partition problem. Then by introducing the notion of “additively prime” we obtain the generating function for this problem. Finally we establish the analyticity of the generating function.     

Galois points for a plane curve in arbitrary characteristic

In 1996, Hisao Yoshihara introduced a new notion in algebraic geometry: a Galois point for a plane curve is a point from which the projection induces a Galois extension of function fields. Yoshihara has established various new approaches to algebraic geometry by using Galois point or generalized notions of it. It is an interesting problem to determine the distribution of Galois points for a given plane curve. In this paper, we survey recent results related to this problem.      

Second order optimality conditions for bilevel set optimization problems

In this paper, we introduce a new notion of augmenting function known as indicator augmenting function to establish a minmax type duality relation, existence of a path of solution converging to optimal value and a zero duality gap relation for a nonconvex primal problem and the corresponding Lagrangian dual problem. We also obtain necessary and sufficient conditions for an exact penalty representation in the framework of indicator augmented Lagrangian.     

Discontinuous control problems for non-convex dynamics and near viability for singularly perturbed control systems

The aim of this paper is to study two classes of discontinuous control problems without any convexity assumption on the dynamics. In the first part we characterize the value function for the Mayer problem and the supremum cost problem using viscosity tools and the notion of ε-viability (near viability). These value functions are given with respect to discontinuous cost functionals. In the second part we obtain results describing the ε-viability (near viability) of singularly perturbed control systems.     

On the Chromatic Number of Graphs

In this paper, the property of a function to be without exceptional family of elements (EFE) is investigated. We show that, for a pseudomonotone operator, the solvability of a complementarity problem is equivalent to the property of the function to be without EFE. Finally, we study the strict feasibility of a complementarity problem making use of the Leray-Schauder alternative and the notion of EFE.     

Pattern search methods for finite minimax problems

In this paper, we propose pattern search methods for finite minimax problems. Due to the nonsmoothness of this class of problems, we convert the original problem into a smooth one by using a smoothing technique based on the exponential penalty function of Kort and Bertsekas, which technique depends on a smoothing parameter that control the approximation to the finite minimax problems. The proposed methods are based on a sampling of the smooth function along a set of suitable search directions and on an updating rule for the step-control parameter. Under suitable conditions, we get the global convergence results despite the fact that pattern search methods do not have explicit information concerning the gradient and consequently are unable to enforce explicitly a notion of sufficient feasible decrease.     

Exceptional Families of Elements,Feasibility and Complementarity

Feasibility is an important property for a complementarity problem. A complementarity problem is solvable if it is feasible and some supplementary assumptions are satisfied. In this paper, we introduce the notion of (, )-exceptional family of elements for a continuous function and we apply this notion to the study of feasibility of nonlinear complementarity problems.     

Vector subspaces and operators with the Stone condition

We introduce the notion of an operating function on a subset of a F{\Phi} -algebra E. Then we use this notion to generalize results from Huijsmans and de Pagter (Proc. Lond. Math. Soc. 48:161–174, 1984) about the connection between vector subspaces and subalgebras of E. In the second part we investigate the analogous problem for operators.     

Exceptional Family of Elements for a Variational Inequality Problem and its Applications

This paper introduces a new concept of exceptional family of elements (abbreviated, exceptional family) for a finite-dimensional nonlinear variational inequality problem. By using this new concept, we establish a general sufficient condition for the existence of a solution to the problem. Such a condition is used to develop several new existence theorems. Among other things, a sufficient and necessary condition for the solvability of pseudo-monotone variational inequality problem is proved. The notion of coercivity of a function and related classical existence theorems for variational inequality are also generalized. Finally, a solution condition for a class of nonlinear complementarity problems with so-called P _* -mappings is also obtained.     

Well-posedness and stability in vector optimization problems using Henig proper efficiency

In this article, we study well-posedness and stability aspects for vector optimization in terms of minimizing sequences defined using the notion of Henig proper efficiency. We justify the importance of set convergence in the study of well-posedness of vector problems by establishing characterization of well-posedness in terms of upper Hausdorff convergence of a minimizing sequence of sets to the set of Henig proper efficient solutions. Under certain compactness assumptions, a convex vector optimization problem is shown to be well-posed. Finally, the stability of vector optimization is discussed by considering a perturbed problem with the objective function being continuous. By assuming the upper semicontinuity of certain set-valued maps associated with the perturbed problem, we establish the upper semicontinuity of the solution map.     

Viscosity solutions of Eikonal equations on topological networks

In this paper we introduce a notion of viscosity solutions for Eikonal equations defined on topological networks. Existence of a solution for the Dirichlet problem is obtained via representation formulas involving a distance function associated to the Hamiltonian. A comparison theorem based on Ishii’s classical argument yields the uniqueness of the solution.     

Application of the finite integral transform method to solving a mixed problem with integro-differential conditions for a nonclassical equation

We consider a mixed problem with integro-differential boundary conditions for a nonclassical equation. Under certain conditions, we apply a finite integral transform to this problem and obtain a parametric problem. We introduce the notion of proper boundary conditions of the parametric problem, which is wider than the notion of regularity. By applying the inverse integral transform to the solution of the parametric problem, we obtain an analytic representation of the solution of the original mixed problem.     

Exceptional Families of Elements for Optimization Problems in Reflexive Banach Spaces with Applications

In this paper, we propose a new notion of ‘exceptional family of elements’ for convex optimization problems. By employing the notion of ‘exceptional family of elements’, we establish some existence results for convex optimization problem in reflexive Banach spaces. We show that the nonexistence of an exceptional family of elements is a sufficient and necessary condition for the solvability of the optimization problem. Furthermore, we establish several equivalent conditions for the solvability of convex optimization problems. As applications, the notion of ‘exceptional family of elements’ for convex optimization problems is applied to the constrained optimization problem and convex quadratic programming problem and some existence results for solutions of these problems are obtained.     

Renormalised solutions in thermo-visco-plasticity for a Norton–Hoff type model. Part II: The limit case

In this article we define a new notion of solutions in thermo-visco-plasticity. Using results from our previous work Chełmiński and Owczarek (2016) we analyse the limit case and prove existence of renormalised solutions to the considered problem assuming that the inelastic constitutive function is of the Norton–Hoff type.     

A derivative concept with respect to an arbitrary kernel and applications to fractional calculus

In this paper, we propose a new concept of derivative with respect to an arbitrary kernel function. Several properties related to this new operator, like inversion rules and integration by parts, are studied. In particular, we introduce the notion of conjugate kernels, which will be useful to guaranty that the proposed derivative operator admits a right inverse. The proposed concept includes as special cases Riemann‐Liouville fractional derivatives, Hadamard fractional derivatives, and many other fractional operators. Moreover, using our concept, new fractional operators involving certain special functions are introduced, and some of their properties are studied. Finally, an existence result for a boundary value problem involving the introduced derivative operator is proved.     

Periodic schedules for linear precedence constraints

We consider the computation of periodic cyclic schedules for linear precedence constraints graphs: a linear precedence constraint is defined between two tasks and induces an infinite set of usual precedence constraints between their executions such that the difference of iterations is a linear function. The objective function is the minimization of the maximal period of a task.We recall first that this problem may be modelled using linear programming. A polynomial algorithm is then developed to solve it for a particular class of linear precedence graphs called unitary graphs. We also show that a periodic schedule may not exist for unitary graphs. In the general case, a decomposition of the linear precedence graph into unitary components is computed and we assume that a periodic schedule exists for each of these components. Lower bounds on the periods are exhibited and we show that an optimal periodic schedule may not achieve them. The notion of quasi-periodic schedule is then introduced and we prove that this new class of schedules always reaches these bounds.     

Image of a parametric optimization problem and continuity of the perturbation function

In this note, we consider the notion of the image of a parametric optimization problem and show that the lower semicontinuity and upper semicontinuity properties of its marginal function can be equivalently expressed as two geometric relations in the image space. These results generalize some existing statements in the literature.     

E-凸条件下多目标规划的Mond-Weir型对偶研究

本文主要研究E-凸函数的若干性质,引入E-凸多目标规划的定义,建立E-凸多目标规划的Mond-Weir型对偶问题,并在E.凸条件假设下,证明E-凸多目标规划的弱对偶性、直接对偶性及逆对偶性.     

Fuzzy optimal control problem with non-linear functional

In this paper, we introduce a new space of fuzzy numbers equipped with a scalar product defined in this space. The notion of a derivative of a fuzzy function in this space is defined. By employing these notions, an optimal control problem with non-linear functional is formulated and an optimality condition is obtained in the form of maximum principle. Using this result, the numerical algorithm is offered for the solution of such problems.