We consider optimization of nonlinear objective functions that balance $d$ linear criteria over $n$-element independence systems presented by linear-optimization oracles. For $d=1$, we have previously shown that an $r$-best approximate solution can be found in polynomial time. Here, using an extended Erdos-Ko-Rado theorem of Frankl, we show that for $d=2$, finding a $\rho n$-best solution … Read more
This paper studies the vector optimization problem of finding weakly ef- ficient points for maps from Rn to Rm, with respect to the partial order induced by a closed, convex, and pointed cone C ⊂ Rm, with nonempty inte- rior. We develop for this problem an extension of the proximal point method for scalar-valued convex … Read more
In 2004, Graña Drummond and Iusem proposed an extension of the projected gradient method for constrained vector optimization problems. In that method, an Armijo-like rule, implemented with a backtracking procedure, was used in order to determine the steplengths. The authors just showed stationarity of all cluster points and, for another version of the algorithm (with … Read more
Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body~$\mathfrak x$, we try to maximize the volume of~$\mathfrak x$ and minimize the width of~$\mathfrak x$ simultaneously. These problems are addressed … Read more
We present a proximal point method to solve multiobjective problems based on the scalarization for maps. We build a family of a convex scalar strict representation of a convex map F with respect to the lexicographic order on Rm and we add a variant of the logarithmquadratic regularization of Auslender, where the unconstrained variables in … Read more
A simple unified framework is presented for the study of strong efficient solutions, weak efficient solutions, positive proper efficient solutions, Henig global proper efficient solutions, Henig proper efficient solutions, super efficient solutions, Benson proper efficient solutions, Hartley proper efficient solutions, Hurwicz proper efficient solutions and Borwein proper efficient solutions of set-valued optimization problem with/or without … Read more
In this paper we present a primal-dual interior-point algorithm to solve a class of multi-objective network flow problems. More precisely, our algorithm is an extension of the single-objective primal-dual infeasible and inexact interior point method for multi-objective linear network flow problems. A comparison with standard interior point methods is provided and experimental results on bi-objective … Read more
In this paper we consider, for the first time, approximate Henig proper minimizers and approximate super minimizers of a set-valued map F with values in a partially ordered vector space and formulate two versions of the Ekeland variational principle for these points involving coderivatives in the senses of Ioffe, Clarke and Mordukhovich. As applications we … Read more
In this paper, we propose a new recombination operator and test its performance in the context of the multiobjective 0/1 knapsack problem (MOKP). The proposed recombination operator generates only one offspring solution from a selected pair of parents according to the two following stages. In the first stage, called genetic shared-information stage or similarity-preserving stage, … Read more
We propose an extension of Newton’s Method for unconstrained multiobjective optimization (multicriteria optimization). The method does not scalarize the original vector optimization problem, i.e. we do not make use of any of the classical techniques that transform a multiobjective problem into a family of standard optimization problems. Neither ordering information nor weighting factors for the … Read more