Exact penalty property for a class of inequality-constrained minimization problems

We use the penalty approach in order to study constrained minimization problems. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish the exact penalty property for a large class of inequality-constrained minimization problems.     

Existence of exact penalty for optimization problems with mixed constraints in Banach spaces

In this paper we use the penalty approach in order to study constrained minimization problems in a Banach space with nonsmooth nonconvex mixed constraints. A penalty function is said to have the exact penalty property [J.-B. Hiriart-Urruty, C. Lemarechal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993] if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish sufficient conditions for the exact penalty property.     

Existence of Exact Penalty for Constrained Optimization Problems in Metric Spaces

In this paper we use the penalty approach in order to study constrained minimization problems in a complete metric space with locally Lipschitzian mixed constraints. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish sufficient conditions for the exact penalty property.      

A lower bound for the controlling parameters of the exact penalty functions

The purpose of this paper is to present new exact penalty functions and discuss their properties. A lower bound on the controlling parameters is given, for which above this value, the optimum of the exact penalty function coincides with the optimum of the nonlinear programming problem.This work was supported by the National Research Council of Canada under Grant A4414.     

Parametrizations of Kojima’s system and relations to penalty and barrier functions

We investigate two homotopies that perturb Kojima’s system for describing critical points of a nonlinear optimization problem in finite dimension. Each of them characterizes stationary points of a usual penalty and a new “barrier” function. The latter is a continuous deformation of the objective, symmetric to the penalty from a formal point of view. Stationary points of these functions appear as perturbed critical points and vice versa. This permits new interpretations of the related solution methods and allows estimates of the solutions by using implicit function theorems for Lipschitzian equations.     

Existence and multiplicity of positive solutions for some Hardy-Sobolev critical elliptic equation with boundary singularities

In this paper, by Ekeland’s variational principle and strong maximum principle, we consider the existence and multiplicity of positive solutions for some semilinear elliptic equation involving critical Hardy-Sobolev exponents and Hardy terms with boundary singularities.     

Variational inclusions problems with applications to Ekeland’s variational principle,fixed point and optimization problems

In this paper, we prove the existence theorems of two types of systems of variational inclusions problem. From these existence results, we establish Ekeland’s variational principle on topological vector space, existence theorems of common fixed point, existence theorems for the semi-infinite problems, mathematical programs with fixed points and equilibrium constraints, and vector mathematical programs with variational inclusions constraints.     

Generalized Invariant Monotonicity and Invexity of Non-differentiable Functions

This paper is devoted to the study of relationships between several kinds of generalized invexity of locally Lipschitz functions and generalized monotonicity of corresponding Clarke’s subdifferentials. In particular, some necessary and sufficient conditions of being a locally Lipschitz function invex, quasiinvex or pseudoinvex are given in terms of momotonicity, quasimonotonicity and pseudomonotonicity of its Clarke’s subdifferential, respectively. As an application of our results, the existence of the solutions of the variational-like inequality problems as well as the mathematical programming problems (MP) is given. Our results extend and unify the well known earlier works of many authors.     

Existence theorems for variational inclusion problems and the set-valued vector Ekeland variational principle in a complete metric space

In this paper, we apply an existence theorem for the variational inclusion problem to study the existence results for the variational intersection problems in Ekeland’s sense and the existence results for some variants of set-valued vector Ekeland variational principles in a complete metric space. Our results contain Ekeland’s variational principle as a special case and our approaches are different to those for any existence theorems for such problems.     

Systems of equilibrium problems with applications to new variants of Ekeland’s variational principle,fixed point theorems and parametric optimization problems

In this paper, we first establish some existence theorems of systems of generalized vector equilibrium problems. From these results, we obtain new variants of Ekeland’s variational principle in a Hausdorff t.v.s., a minimax theorem and minimization theorems. Some applications to the existence theorem of systems of semi-infinite problem, a variant of flower petal theorem and a generalization of Schauder’s fixed point theorem are also given.     

The exact number of solutions for the second order nonlinear boundary value problem

A second order nonlinear differential equation with homogeneous Dirichlet boundary conditions is considered. An explicit expression for the root functions for an autonomous nonlinear boundary value problem is obtained using the results of the paper [SOMORA, P.: The lower bound of the number of solutions for the second order nonlinear boundary value problem via the root functions method, Math. Slovaca 57 (2007), 141–156]. Other assumptions are supposed to prove the monotonicity of root functions and to get the exact number of solutions. The existence of infinitely many solutions of the boundary value problem with strong nonlinearity is obtained by the root function method as well. The paper was supported by the Grant VEGA No. 2/7140/27, Bratislava.     

A penalty optimization technique for a class of regulator problems

This paper describes a function space algorithm for the solution of a class of linear-quadratic optimal control problems.The research of the first author was supported by Senate Research Grant No. 8.187.17, University of Ilorin, Ilorin, Kwara State, Nigeria.The authors thank two anonymous referees for their useful and challenging suggestions and comments which improved the quality of this paper. The authors are also indebted to I. Orisamolu for carrying out the computing work.     

Existence and Concentration of Ground States of Coupled Nonlinear Schr¨odinger Equations with Bounded Potentials

A 2-coupled nonlinear Schrbdinger equations with bounded varying potentials and strongly attractive interactions is considered. When the attractive interaction is strong enough, the existence of a ground state for sufficiently small Planck constant is proved. As the Planck constant approaches zero, it is proved that one of the components concentrates at a minimum point of the ground state energy function which is defined in Section 4.     

A necessary and sufficient condition for duality in multiobjective variational problems

In this paper we move forward in the study of duality and efficiency in multiobjective variational problems. We introduce new classes of pseudoinvex functions, and prove that not only it is a sufficient condition to establish duality results, but it is also necessary. Moreover, these functions are characterized in order that all Kuhn–Tucker or Fritz John points are efficient solutions. Recent papers are improved. We provide an example to show this improvement and illustrate these classes of functions and results.     

Quasi-variational relation problems and generalized Ekeland's variational principle with applications

In this article, we study the existence of solutions for a quasivariational relation problem and then give applications to the existence of solutions for set-valued Ekeland's principle, generalized vector Ekeland's variational principle and generalized equilibrium problems. Our results and techniques of proof are different from any existence result in the literature.     

Comparison of discrete mixed means containing symmetric functions

We prove inequalities for mixed generalized means; from these inequalities we obtain analogs or generalizations of inequalities due to Ness, to Carlson, Meany, and Nelson, to Godunova, and to Marcus and Minc, etc.     

On the Gerber-Shiu discounted penalty function for subexponential claims

We obtain the asymptotics of the Gerber-Shiu discounted penalty function in the classical Lundberg model. We cosider claims from a class of subexponential distributions and find the asymptotics as the initial surplus x tends to infinity. The main term of the discounted penalty function ψ(x, δ) has different expressions in the cases where the interest rate δ > 0 and where δ = 0. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 598–605, October–December, 2006.     

An Ekeland’s variational principle for set-valued mappings with applications

In this paper, we obtain a general Ekeland’s variational principle for set-valued mappings in complete metric space, which is different from those in [G.Y. Chen, X.X. Huang, Ekeland’s ε-variational principle for set-valued mapping, Mathematical Methods of Operations Research 48 (1998) 181–186; G.Y. Chen, X.X. Huang, S.H. Hou, General Ekeland’s Variational Principle for Set-Valued Mappings, Journal of Optimization Theory and Applications 106 (2000) 151–164; S.J. Li, W.Y. Zhang, On Ekeland’s variational Principle for set-valued mappings, Acta Mathematicae Application Sinica, English Series 23 (2007) 141–148]. By the result, we prove some existence results for a general vector equilibrium problem under nonconvex and compact or noncompact assumptions of its domain, respectively. Moreover, we give some equivalent results to the variational principle.     

ON A CLASS OF BESICOVITCHFUNCTIONS TO HAVE EXACT BOX DIMENSION： A NECESSARY AND SUFFICIENT CONDITION

This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.