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In the present paper, we analyze a class of convex semi-infinite programming problems with arbitrary index sets defined by a finite number of nonlinear inequalities. The analysis is carried out by employing the constructive approach, which, in turn, relies on the notions of immobile indices and their immobility orders. Our previous work showcasing this approach includes a number of papers dealing with simpler cases of semi-infinite problems than the ones under consideration here. Key findings of the paper include the formulation and the proof of implicit and explicit optimality conditions under assumptions, which are less restrictive than the constraint qualifications traditionally used. In this perspective, the optimality conditions in question are also compared to those provided in the relevant literature. Finally, the way to formulate the obtained optimality conditions is demonstrated by applying the results of the paper to some special cases of the convex semi-infinite problems. [ABSTRACT FROM AUTHOR]

In this paper a new hybrid conjugate gradient algorithm is proposed and analyzed. The parameter β k is computed as a convex combination of the Polak-Ribière-Polyak and the Dai-Yuan conjugate gradient algorithms, i.e. β=(1− θ k) β+ θ k β. The parameter θ k in the convex combination is computed in such a way that the conjugacy condition is satisfied, independently of the line search. The line search uses the standard Wolfe conditions. The algorithm generates descent directions and when the iterates jam the directions satisfy the sufficient descent condition. Numerical comparisons with conjugate gradient algorithms using a set of 750 unconstrained optimization problems, some of them from the CUTE library, show that this hybrid computational scheme outperforms the known hybrid conjugate gradient algorithms. [ABSTRACT FROM AUTHOR]

The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems. [ABSTRACT FROM AUTHOR]

We consider a convexification method for a class of nonsmooth monotone functions. Specifically, we prove that a semismooth monotone function can be converted into a convex function via certain convexification transformations. The results derived in this paper lay a theoretical base to extend the reach of convexification methods in monotone optimization to nonsmooth situations. [ABSTRACT FROM AUTHOR]