The method of periodic projections consists in iterating projections onto $m$ closed convex subsets of a Hilbert space according to a periodic sweeping strategy. In the presence of $m\geq 3$ sets, a long-standing question going back to the 1960s is whether the limit cycles obtained by such a process can be characterized as the minimizers … Read more
Continuous linear programs have attracted considerable interest due to their potential for modelling manufacturing, scheduling and routing problems. While efficient simplex-type algorithms have been developed for separated continuous linear programs, crude time discretization remains the method of choice for solving general (non-separated) problem instances. In this paper we propose a more generic approximation scheme for … Read more
The Kantorovich function $(x^TAx)( x^T A^{-1} x)$, where $A$ is a positive definite matrix, is not convex in general. From a matrix or convex analysis point of view, it is interesting to address the question: When is this function convex? In this paper, we prove that the 2-dimensional Kantorovich function is convex if and only … Read more
In this paper, we propose an efficient approach for solving a class of convex optimization problems in Hilbert spaces. Our feasible region is a (possibly infinite-dimensional) simple convex set, i.e. we assume that projections on this set are computationally easy to compute. The problem we consider is the minimization of a convex function over this … Read more
In this paper new results are established in multiobjective DC programming with infinite convex constraints ($MOPIC$ for abbr.) that are defined on Banach space (finite or infinite) with objectives given as the difference of convex functions subject to infinite convex constraints. This problem can also be called multiobjective DC semi-infinite and infinite programming, where decision … Read more
We introduce non-autonomous continuous dynamical systems which are linked to Newton and Levenberg-Marquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on Minty representation of maximal monotone operators as lipschitzian manifolds, we show that these dynamics can be formulated as first-order in time differential systems, which are relevant … Read more
Many physical phenomena, governed by partial differential equations (PDEs), are second order in nature. This makes sense to pose the control on the second order derivatives of the field solution, in addition to zero and first order ones, to consistently control the underlaying process. However, this type of control is nontrivial and to the best … Read more
The automatic optimal design of feeding system in the shape casting process is considered in the present work. In fact, the goal is to find the optimal position, size, shape and topology of risers in the shape casting process. This problem is formulated as a minimum weight topology optimization problem subjected to a nonlinear transient … Read more
Following [13] we provide new formulas for the Fenchel subdifferential of the conjugate of functions defined on locally convex spaces. In particular, this allows deriving expressions for the minimizers set of the lower semicontinuous convex hull of such functions. These formulas are written by means of primal objects related to the subdifferential of the initial … Read more
We solve a class of convex infinite-dimensional optimization problems using a numerical approximation method that does not rely on discretization. Instead, we restrict the decision variable to a sequence of finite-dimensional linear subspaces of the original infinite-dimensional space and solve the corresponding finite-dimensional problems in a efficient way using structured convex optimization techniques. We prove … Read more