It is a challenging task to identify the objectives on which a certain decision was based, in particular if several, potentially conflicting criteria are equally important and a continuous set of optimal compromise decisions exists. This task can be understood as the inverse problem of...

Recently, Kronqvist et al. (J Global Optim 64(2):249–272, 2016) rediscovered the supporting hyperplane algorithm of Veinott (Oper Res 15(1):147–152, 1967) and demonstrated its computational benefits for solving convex mixed integer nonlinear programs. In this paper we derive the algorithm...

An important aspect of optimization algorithms, for instance evolutionary algorithms, are termination criteria that measure the proximity of the found solution to the optimal solution set. A frequently used approach is the numerical verification of necessary optimality conditions such as the...

In this paper we provide a generalization of a Positivstellensatz by Pólya [Pólya in Naturforsch Ges Zürich 73:141–145 <CitationRef CitationID="CR18">1928</CitationRef>]. We show that if a homogeneous polynomial is positive over the intersection of the non-negative orthant and a given basic semialgebraic cone (excluding the origin),...</citationref>

In the present paper, we are devoted to exploring conditions of well-posedness for hemivariational inequalities in reflexive Banach spaces. By using some equivalent formulations of the hemivariational inequality considered under different monotonicity assumptions, we establish two kinds of...

In this paper we adapt the main results from Amahroq and Gadhi (J Glob Optim 21:435–443, <CitationRef CitationID="CR2">2001</CitationRef>) for a general set-valued optimization problem to an optimistic bilevel programming problem as an optimization problem with implicitly given set-valued constraint. Since this constraint is assumed to...</citationref>

Multi-objective optimization problems are often solved by a sequence of parametric single-objective problems, so-called scalarizations. If the set of nondominated points is finite, the entire nondominated set can be generated in this way. In the bicriteria case it is well known that this can be...

In the paper, we introduce the concepts of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$G$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>G</mi> </math> </EquationSource> </InlineEquation>-type I and generalized <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$G$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>G</mi> </math> </EquationSource> </InlineEquation>-type I functions for a new class of nonconvex multiobjective variational control problems. For such nonconvex vector optimization problems, we prove sufficient optimality conditions for weakly...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>

In many engineering optimization problems, objective function evaluations can be extremely computationally expensive. The effective global optimization (EGO) is a widely used approach for expensive optimization. Balance between global exploration and local exploitation is a very important issue...