We consider the DC optimization problem (P) inf x∈X {(f1(x) − f2(x)) + (g1(Ax) − g2(Ax))}, where f1, f2, g1 and g2 are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y respectively, and A is a linear continuous operator from X to Y . Adopting the standard convexification technique, a Fenchel dual problem of (P) is given. By using properties of the ep...

Many high dimensional data mining problems can be formulated as minimizing an empirical loss function with a penalty proportional to the number of variables required to describe a model. We propose a graduated non-convexification method to facilitate tracking of a global minimizer of this problem. We prove that under some conditions the proposed regularization problem using the continuous piece...

We analyze in this work a spatio-temporal optimal design problem governed by a linear damped 1-D wave equation. The problem consists in seeking simultaneously the spatiotemporal layout of two isotropic materials and the static position of the damping set in order to minimize a functional depending quadratically on the gradient of the state. The lack of classical solutions for this kind of nonli...

In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using sequen...

This paper introduces in a natural way notion of horizontal convex envelopes continuous functions the Heisenberg group. We provide convexification process to find envelope constructive manner. also apply show h-convexity viscosity solutions class fully nonlinear elliptic equations group satisfying certain symmetry condition. Our examples that general one cannot expect without