We present necessary and sufficient conditions for reachable sets of discrete-time systems x(k+1) = F (k, x(k)) to be convex. In particular, the set of states reachable at a given time from a sufficiently small ellipsoid of initial states is always convex if F is smooth enough, and we provide explicit bounds on the size of those ellipsoids. Our results imply that outer discrete approximations w...

The paper deals with solutions of a differential inclusion ẋ ∈ F (x) constrained to a compact convex set Ω. Here F is a compact, possibly non-convex valued, Lipschitz continuous multifunction, whose convex closure coF satisfies a strict inward pointing condition at every boundary point x ∈ ∂Ω. Given a reference trajectory x∗(·) taking values in an ε-neighborhood of Ω, we prove the existence of ...

We consider the convex optimization problem P : minx{f(x) : x ∈ K} where f is convex continuously differentiable, and K ⊂ R is a compact convex set with representation {x ∈ R : gj(x) ≥ 0, j = 1, . . . ,m} for some continuously differentiable functions (gj). We discuss the case where the gj ’s are not all concave (in contrast with convex programming where they all are). In particular, even if th...

Let F be a convex n-gon in a horizontal plane of the Euclidean 3-space. Consider its spatial variation under which its vertices move vertically and let F be the convex hull of such a variation. In the general position, the boundary of F splits naturally into the "bottom" F 0 and the "top" F 00. The polyhedron F 0 (F 00) has triangular faces and no vertices inside. The projection of these faces ...

We study the minimization of a convex function f(X) over the set of n × n positive semi-definite matrices, but when the problem is recast as minU g(U) := f(UU >), with U ∈ Rn×r and r ≤ n. We study the performance of gradient descent on g—which we refer to as Factored Gradient Descent (Fgd)—under standard assumptions on the original function f . We provide a rule for selecting the step size and,...

We prove the existence of viable solutions to the Cauchy problem x′′ ∈ F (x, x′), x(0) = x0, x′(0) = y0, where F is a set-valued map defined on a locally compact set M ⊂ R, contained in the Fréchet subdifferential of a φ-convex function of order two.

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.

We consider the constrained optimization problem  defined by:
 $$f (x^*) = \min_{x \in  X} f(x)\eqno (1)$$
 
 where function  f : \pmb{\mathbb{R}}^{n} → \pmb{\mathbb{R}} is convex  on a closed bounded convex set X.
 To solve problem (1), most methods transform this into without constraints, either by introducing Lagrange multiplie...

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...

A generalized semitoric system F := (J,H) : M → R on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S-action and is not necessarily proper. These systems can exhibit focusfocus singularities, which correspond to fibers of F which are topologically multi-pinched tori. The image F (M) is...