Bulletin of Iranian Mathematical Society
Volume:42 Issue: 7, 2016

We consider nonconvex vector optimization problems with variable ordering structures in Banach spacesý. ýUnder certain boundedness and continuity properties we present necessary conditions for approximate solutions of these problemsý. ýUsing a generic approach to subdifferentials we derive necessary conditions for approximate minimizers and approximately minimal solutions of vector optimization problems with variable ordering structures applying nonlinear separating functionals and Ekeland's variational principleý.

We analyze a sequential decision making process, in which at each step the decision is made in two stages. In the rst stage a partially optimal action is chosen, which allows the decision maker to learn how to improve it under the new environment. We show how inertia (cost of changing) may lead the process to converge to a routine where no further changes are made. We illustrate our scheme with some economic models.

We study the support sets of sub-topical functionsý ýand investigate their maximal elements in order to establish a necessary and sufficient conditioný ýfor the global minimum of the difference of two sub-topical functionsý.

We consider a linear programming problem in a general form and suppose that all coefficients may vary in some prescribed intervalsý. ýContrary to classical modelsý, ýwhere parameters can attain any value from the interval domains independentlyý, ýwe study problems with linear dependencies between the parametersý. ýWe present a class of problems that are easily solved by reduction to the classical caseý. ýIn contrastý, ýwe also show a class of problems with very simple dependenciesý, ýwhich appear to be hard to deal withý. ýWe also point out some interesting open problemsý.

We present an improved version of a full Nesterov-Todd step infeasible interior-point method for linear complementarity problem over symmetric cone (Bull. Iranian Math. Soc., 40(3), 541-564, (2014)). In the earlier version, each iteration consisted of one so-called feasibility step and a few -at most three - centering steps. Here, each iteration consists of only a feasibility step. Thus, the new algorithm demands less work in each iteration and admits a simple analysis of complexity bound. The complexity result coincides with the best-known iteration bound for infeasible interior-point methods.

Hereý, ýwe aim to develop a new algorithm for solving a multiobjective linear programming problemý. ýThe algorithm is to obtain a solution which approximately meets the decision maker's preferencesý. ýIt is proved that the proposed algorithm always converges to a weak efficient solution and at times converges to an efficient solutioný. ýNumerical examples and a simulation study are used to illustrate the performance of the proposed algorithmý.

A common approach to determine efficient solutions of a multiple objective optimization problemý ýis reformulating it to a parameter dependent scalar optimization problemý. ýThis reformulation is called scalarization approachý. Here, a well-known scalarization approach named Pascoletti-Serafini scalarization is consideredý. First, some difficulties of this scalarization are discussed and then removed by restricting the parameter setý. A method is presented to convert a space ordered by a specificý ýordering cone to an equivalent space ordered by the natural ordering coneý. ýUtilizing the presented conversioný, ýall confirmed results and theorems forý ýmultiple objective optimization problems ordered by the naturalý ýordering cone can be extended to multiple objective optimizationý ýproblems ordered by specific ordering cones.