Issue 3, Volume 26, December 2001

2001 · Open access

1. Foreword PDF
2. ON NONCONVEX OPTIMIZATION B. D. CRAVEN PDF

   ▸ Abstract

   Constrained minimization problems, in particular optimal control problems, are considered, where convexity requirements may not be satisfied. Various criteria for existence, or uniqueness, of a minimum are discussed. Criteria for necessary (Karush-Kuhn-Tucker or Pontryagin) conditions to be also sufficient for a minimum are discussed as well. Some of these depend on generalized-convexity properties such as invexity or V-invexity.

   249-256
3. SECOND-ORDER OPTIMALITY CONDITIONS FOR $C^1$ MULTIOBJECTIVE PROGRAMMING PROBLEMS A. GUERRAGGIO, D. T. LUC, N. B. MINH PDF

   ▸ Abstract

   In this paper we use approximate Hessian matrices of continuously differentiable vector functions to establish second order optimality conditions for constrained multiobjective programming problems with data of class $C^1$.

   257-268
4. BOUNDEDNESS OF SYMMETRICALLY $\gamma$-CONVEX FUNCTIONS NGUYEN NGOC HAI, HOANG XUAN PHU PDF

   ▸ Abstract

   A function $f: D\to\mathbb R$ is said to be symmetrically $\gamma$-convex w.r.t. the roughness degree $\gamma > 0$ if the Jensen inequality $$f(x_{\lambda})\leq (1-\lambda)f(x_0)+\lambda f(x_1), \quad x_{\lambda}:=(1-\lambda)x_0+\lambda x_1$$ is fulfilled for all $x_0, x_1 \in D$ satisfying $\|x_0-x_1\|\geq \gamma$ and for $$\lambda=\dfrac{\gamma}{\|x_1-x_0\|}\quad \text{and}\quad \lambda=1-\dfrac{\gamma}{\|x_1-x_0\|}.$$ Such a function also has some analytical properties which are similar to those of convex functions. For instance, if it is bounded above on some sphere $\{x\in X\,:\, \|x-x^*\|=\gamma\}\subset D$ then it is bounded on the ball $\overline{\mathcal{U}}_{\gamma}(x^*):=\{x\in X\,:\, \|x-x^*\|\leq \gamma\}$ and bounded below on each bounded subset of $D$. If the domain $D$ is so large that its interior contains some ball $\overline{\mathcal{U}}_{\gamma}(x^*)$, and if the symmetrically $\gamma$-convex function considered is locally bounded above at some interior point of $D$, then it is locally bounded in the interior of $D$.

   269-277
5. NECESSARY OPTIMALITY CONDITIONS IN PROBLEMS INVOLVING SET-VALUED MAPS WITH PARAMETERS PHAN QUOC KHANH, LE MINH LUU PDF

   ▸ Abstract

   The Fritz John and Kuhn-Tucker necessary optimality conditions are proved for problem $\min F(x,u)$, s.t. $G(x,u)\subset -M$, $0\in P(x,u)$ or s.t. $G(x,u)\cap (-M)\not=\emptyset$, $0\in P(x,u)$, where $x$ is the state variable, $u$ is a parameter, $F$, $G$ and $P$ are multifunctions.

   279-295
6. A COUNTEREXAMPLE ON THE CLOSEDNESS OF THE CONVEX HULL OF A CLOSED CONE IN $R^n$ G. M. LEE, J. Y. SHIN, H. S. SIM PDF

   ▸ Abstract

   In this note, the following question, not discussed well in the literature, is considered: $$ \text{Is the convex hull of } A \text{ closed if } A \text{ is a closed cone in } R^n?$$ When $n = 1$ or $2$, the answer to the above problem is affirmative. The main purpose of this brief note is to give a simple example showing that the answer is negative when $n\geq 3$.

   279-299
7. ON PROPER EFFICIENCIES IN LOCALLY CONVEX SPACES–A SURVEY JING LIU, WEN SONG PDF

   ▸ Abstract

   In this paper, we consider the main definitions of proper efficiency in locally convex topological vector spaces and examine the relationships among them.

   301-312
8. INCREASING POSITIVELY HOMOGENEOUS FUNCTIONS DEFINED ON $\mathbb R^n$ J. E. MARTINEZ-LEGAZ, A. M. RUBINOV PDF

   ▸ Abstract

   The theory of IPH (increasing positively homogeneous of degree one) functions defined on the cone $\mathbb R^n_+$ of all vectors with nonnegative coordinates is well developed. In this paper we present a suitable extension of this theory for IPH functions defined on the entire space $\mathbb R^n$.

   313-331
9. METHODS FOR FINDING GLOBAL OPTIMAL SOLUTIONS TO LINEAR PROGRAMS WITH EQUILIBRIUM CONSTRAINTS LE DUNG MUU, NGUYEN VAN QUY PDF

   ▸ Abstract

   A mathematical program with equilibrium constraints is an optimization problem with two sets of variables $x$ and $y$, in which some or all of its constraints are defined by a parametric variational inequality or complementarity system with $y$ as its primary variables and $x$ the parameter vector. This problem even in the linear case is a difficult global optimization problem because of its nested structure. We use a dual formulation of this problem to develop two decomposition branch-and-bound algorithms for finding a global optimal solution of a linear mathematical program with equilibrium constraints. The first algorithm uses a simplicial subdivision whereas the second one uses a binary tree defined by using the sign (zero or positive) of the dual variables. The searching trees in both lgorithms are created in the dual space. Preliminary computational experiences and results show that on a PC computer the lgorithm using binary tree can solve linear problems with equilibrium constraints up to twenty-five dual variables; the number of the primal variables may be larger.

   333-347
10. SOME EQUILIBRIUM PROBLEMS IN GENERALIZED CONVEX SPACES SEHIE PARK PDF

    ▸ Abstract

    We show that how the fundamental theorems on equilibrium problems can be extended to generalized convex spaces. Precisely, most of important results in the KKM theory hold without assuming the linearity in topological vector spaces. Such examples are the KKM theorem, the minimax theorem and the intersection lemma of von Neumann, the Nash equilibrium theorem, various fixed point theorems, Ky Fan’s minimax inequality, variational inequalities, best approximation theorems, existence theorems for solutions of generalized quasi-equilibrium problems, and others.

    349-364
11. MEAN VALUE THEOREMS FOR CORRESPONDENCES JEAN-PAUL PENOT PDF

    ▸ Abstract

    We prove mean value estimates yielding Lipschitz rate for multimappings using coderivatives and subdifferential calculus.

    365-376
12. DIRECTIONAL DIFFERENTIABILITY OF THE OPTIMAL VALUE FUNCTION IN INDEFINITE QUADRATIC PROGRAMMING NGUYEN NANG TAM PDF

    ▸ Abstract

    We obtain an explicit formula for computing the directional derivative of the optimal value function in a general parametric quadratic programming problem with linear constraints. Our result can be used in some cases where the existing results on differential stability in nonlinear programming (applied to quadratic programming) cannot be used.

    377-394
13. ON EXPLICIT VISCOSITY SOLUTIONS TO NONCONVEX-NONCONCAVE HAMILTON-JACOBI EQUATIONS TRAN DUC VAN, MAI DUC THANH PDF

    ▸ Abstract

    We consider the Cauchy problem for Hamilton-Jacobi equations in the case where the Hamiltonian is supposed to be a sum of a convex and a concave function and to depend also on the unknown function. Hopf-Oleinik-Lax-type formulas for viscosity sub- and super-solutions are presented. A sharp estimate for the unique viscosity solution is established.

    395-405
14. LINEAR OPERATORS SATISFYING THE ASSUMPTIONS OF SOME GENERALIZED LAX-MILGRAM THEOREMS NGUYEN DONG YEN, BUI TRONG KIM PDF

    ▸ Abstract

    This paper analyzes the structure of two classes of linear operators satisfying the assumptions of two generalized Lax-Milgram theorems of J. Saint Raymond. For the first class, complete characterizations are proposed. For the second class, some preliminary results are shown. In particular, we prove that the second class is strictly larger than the first one which, in its turn, is strictly larger than the class of elliptic operators.

    407-417