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We establish differentiability properties of the value function of problems of Static Optimization in an abstract infinite dimensional setting and we apply that to problems of Calculus of Variations. We lighten the assumptions of existing…
The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main…
In this paper we study the right differentiability of a parametric infimum function over a parametric set defined by equality constraints. We present a new theorem with sufficient conditions for the right differentiability with respect to…
We consider spatial voting where candidates are located in the Euclidean $d$-dimensional space, and each voter ranks candidates based on their distance from the voter's ideal point. We explore the case where information about the location…
We study an optimal stopping problem with an unbounded, time-dependent and discontinuous reward function. This problem is motivated by the pricing of a variable annuity contract with guaranteed minimum maturity benefit, under the assumption…
We consider the space of real-valued continuously differentiable functions on a compact subset of a euclidean space. We characterize the completeness of this space and prove that the space of restrictions of continuously differentiable…
In this paper we study a problem of looking for an optimal solution of a system of the differential equations with a control and an optimized function. The system of differential equations is changed for two systems with the upper and lower…
We present a new alternative theorems for sequences of functions. As applications, we extend recent results in the literature related to first-order necessary conditions for optimality problems. Our contributions involve extending…
The paper contains a very simple proof of the classical Hasumi's theorem that each usco mapping defined on an extremally disconnected space has a continuous selection. The paper also contains a very simple proof of a recent result about…
We explicitly solve the optimal switching problem for one-dimensional diffusions by directly employing the dynamic programming principle and the excessive characterization of the value function. The shape of the value function and the…
In the Euclidean setting, the well-known Alexandrov theorem states that convex functions are twice differentiable almost everywhere. In this note, we extend this theorem to rank-one convex functions. Our approach is novel in that it draws…
Euclidean preferences are a widely studied preference model, in which decision makers and alternatives are embedded in d-dimensional Euclidean space. Decision makers prefer those alternatives closer to them. This model, also known as…
This paper investigates continuity properties of value functions and solutions for parametric optimization problems. These problems are important in operations research, control, and economics because optimality equations are their…
The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…
Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other…
Along the optimal trajectory of an optimal control problem constrained by a semilinear parabolic partial differential equation, we prove the differentiability of the value function with respect to the initial condition and, under additional…
As a complement to two recent papers by An and Yen [An, D.T.V., Yen, N.D.: Differential stability of convex optimization problems under inclusion constraints. Appl. Anal., 94, 108--128 (2015)], and by An and Yao [An, D.T.V., Yao, J.-C.:…
We provide new results of first-order necessary conditions of optimality problem in the form of John's theorem and in the form of Karush-Kuhn-Tucker's theorem. We establish our result in a topological vector space for problems with…
We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. Firstly, we provide an upper estimate of its Dini--Hadamard subdifferential in terms of the Clarke subdifferential of the…
Differential stability of convex discrete optimal control problems in Banach spaces is studied in this paper. By using some recent results of An and Yen [Appl. Anal. 94, 108--128 (2015)] on differential stability of parametric convex…