First and second order optimality conditions for optimal control problems of state constrained integral equations

This paper deals with optimal control problems of integral equations, with initial-final and running state constraints. The order of a running state constraint is defined in the setting of integral dynamics, and we work here with constraints of arbitrary high orders. First and second-order necessary conditions of optimality are obtained, as well as second-order sufficient conditions

Measuring the balance space sensitivity in vector optimization

Recent literature has shown that the balance space approach may be a significant a1ternative to address several topics concerning vector optimization. Although this new look also leads lo the eflicient set and, consequently, is equivalent to the classical viewpoint, it yields new results and a1gorithms, as well as new economic interpretations, that may be very useful in theoretical framevorks and practical applications. The present paper focuses on the sensitivity of The balance set. We prove a general envelope theorem that yields the sensitivity with respect to any parameter considered in the problem. Fulthermore, we provide a dual problem that characlerizes the primal balance space and its sensitivity. Finally, we a1so give the implications of our results with respect to the sensitivity of the efficient set

Sensitivity in Multiobjective Programming by Differential Equations Methods. The Case of Homogeneous Functions

Proceedings of the Second International Conference on Multi-Objective Programming and Goal Programming, Torremolinos, Spain, May 16-18, 1996.The purpose of this paper is to characterize for convex multiobjective programming, the situations in which the sensitivity with respect to the right side vector of the constraints can be obtained as a solution of a dual program.Publicad

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

Recent progress in random metric theory and its applications to conditional risk measures

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measures. Section 2 gives the central framework in random metric theory, topological structures, important examples, the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals. Section 3 gives several important representation theorems for random conjugate spaces. Section 4 gives characterizations for a complete random normed module to be random reflexive. Section 5 gives hyperplane separation theorems currently available in random locally convex modules. Section 6 gives the theory of random duality with respect to the locally $L^{0}-$convex topology and in particular a characterization for a locally $L^{0}-$convex module to be $L^{0}-$pre$-$barreled. Section 7 gives some basic results on $L^{0}-$convex analysis together with some applications to conditional risk measures. Finally, Section 8 is devoted to extensions of conditional convex risk measures, which shows that every representable $L^{\infty}-$type of conditional convex risk measure and every continuous $L^{p}-$type of convex conditional risk measure ($1\leq p<+\infty$) can be extended to an $L^{\infty}_{\cal F}({\cal E})-$type of $\sigma_{\epsilon,\lambda}(L^{\infty}_{\cal F}({\cal E}), L^{1}_{\cal F}({\cal E}))-$lower semicontinuous conditional convex risk measure and an $L^{p}_{\cal F}({\cal E})-$type of ${\cal T}_{\epsilon,\lambda}-$continuous conditional convex risk measure ($1\leq p<+\infty$), respectively.Comment: 37 page

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models.

We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost based on a Huber-regularised TV seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a combined quasi-Newton/semismooth Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between TGV 2 and ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.King Abdullah University of Science and Technology (KAUST) (Grant ID: KUKI1-007-43), Engineering and Physical Sciences Research Council (Grant IDs: Nr. EP/J009539/1 “Sparse & Higher-order Image Restoration” and Nr. EP/M00483X/1 “Efficient computational tools for inverse imaging problems”), Escuela Politécnica Nacional de Quito (Grant ID: PIS 12-14, MATHAmSud project SOCDE “Sparse Optimal Control of Differential Equations”), Leverhulme Trust (project on “Breaking the non-convexity barrier”), SENESCYT (Ecuadorian Ministry of Higher Education, Science, Technology and Innovation) (Prometeo Fellowship)This is the final version of the article. It first appeared from Springer via http://dx.doi.org/10.1007/s10851-016-0662-