in this paper, we first present a new important property for bouligand tangent cone (contingent cone) of a star-shaped set. we then establish optimality conditions for pareto minima and proper ideal efficiencies in nonsmooth vector optimization problems by means of bouligand tangent cone of image set, where the objective is generalized cone convex set-valued map, in general real normed spaces.

There are two main approaches to binary classification problems: the loss function approach and the uncertainty set approach. The loss function approach is widely used in real-world data analysis. Statistical decision theory has been used to elucidate its properties such as statistical consistency. Conditional probabilities can also be estimated by using the minimum solution of the loss functio...

In this paper, we examine the duality gap between the robust counterpart of a primal uncertain convex optimization problem and the optimistic counterpart of its uncertain Lagrangian dual and identify the classes of uncertain problems which do not have a duality gap. The absence of a duality gap (or equivalently zero duality gap) means that the primal worst value equals the dual best value. We f...

Optimal uncertainty quantification (OUQ) is a framework for numerical extreme-case analysis of stochastic systems with imperfect knowledge of the underlying probability distribution. This paper presents sufficient conditions under which an OUQ problem can be reformulated as a finite-dimensional convex optimization problem, for which efficient numerical solutions can be obtained. The sufficient ...

In this note, we provide a solution to a class of finite-horizon min–max control problems for uncertain max-plus-linear systems where the uncertain parameters are assumed to lie in a given convex and compact set, and it is required that the closed-loop input and state sequence satisfy a given set of linear inequality constraints for all admissible uncertainty realizations. We provide sufficient...

In this paper, we first present a new important property for Bouligand tangent cone (contingent cone) of a star-shaped set. We then establish optimality conditions for Pareto minima and proper ideal efficiencies in nonsmooth vector optimization problems by means of Bouligand tangent cone of image set, where the objective is generalized cone convex set-valued map, in general real normed spaces.

In the present work, we consider Zuckerberg's method for geometric convex-hull proofs introduced in [Geometric convex hull defining formulations, Operations Research Letters 44(5), 625-629 (2016)]. It has only been scarcely adopted literature so far, despite great flexibility designing algorithmic completeness of polyhedral descriptions that it offers. We suspect this is partly due to rather he...

The optimal reactive power dispatch (ORPD) is a very important problem aspect of power system planning and is a highly nonlinear, non-convex optimization problem because consist of both continuous and discrete control variables. Since the power system has inherent uncertainty, hereby, this paper presents both of the deterministic and stochastic models for ORPD problem in multi objective and sin...