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International audience; This paper addresses a large class of vector optimization problems in infinite-dimensional spaces with respect to two important binary relations derived from domination structures. Motivated by theoretical challenges as well as by applications to some models in behavioral sciences, we establish new variational principles that can be viewed as far-going extensions of the Ekeland variational principle to cover domination vector settings. Our approach combines advantages of both primal and dual techniques in variational analysis with providing useful sufficient conditions for the existence of variational traps in behavioral science models with variable domination structures.

Subgradient and Newton algorithms for nonsmooth optimization require generalized derivatives to satisfy subtle approximation properties: conservativity for the former and semismoothness for the latter. Though these two properties originate in entirely different contexts, we show that in the semi-algebraic setting they are equivalent. Both properties for a generalized derivative simply require it to coincide with the standard directional derivative on the tangent spaces of some partition of the domain into smooth manifolds. An appealing byproduct is a new short proof that semi-algebraic maps are semismooth relative to the Clarke Jacobian., 12 pages

In this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously been explored for problems without state constraints. This paper establishes such links in the presence of state constraints and of an additional ordinary control, for locally Lipschitz continuous data.

In this paper, we study the mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with a parametric generalized equation involving the regular normal cone. We derive a new necessary optimality condition which is sharper than the usual M-stationary condition and is applicable even when no constraint qualifications hold for the corresponding mathematical program with complementarity constraints (MPCC) reformulation.

We give criteria for weak and strong invariant closed sets for differential inclusions given in $\mathbb{R}^{n}$ and governed by Lipschitz Cusco perturbations of maximal monotone operators. Correspondingly, we provide different characterizations for the\ associated strong Lyapunov functions. The resulting conditions only depend on the data of the system.

We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed Dirichlet problems. The generalized derivatives can be used to obtain a novel necessary optimality condition for the optimal control of the obstacle problem with control constraints. A comparison shows that this system is stronger than the known system of C-stationarity.

This paper investigates new first-order optimality conditions for general optimization problems. These optimality conditions are stronger than the commonly used M-stationarity conditions and are in particular useful when the latter cannot be applied because the underlying limiting normal cone cannot be computed effectively. We apply our optimality conditions to a MPEC to demonstrate their practicability.

In this paper, an upper semismooth function is defined to be a lower semicontinuous function whose radial subderivative satisfies a mild directional upper semicontinuity property. Examples of upper semismooth functions are the proper lower semicontinuous convex functions, the lower-C$^1$ functions, the regular directionally Lipschitz functions, the Mifflin semismooth functions, the Thibault-Zagrodny directionally stable functions. It is shown that the radial subderivative of such functions can be recovered from any subdifferential of the function. It is also shown that these functions are subdifferentially determined, in the sense that if two functions have the same subdifferential and one of the functions is upper semismooth, then the two functions are equal up to an additive constant., 16 pages

We provide dual sufficient conditions for subtransversality of collections of sets in an Asplund space setting. For the convex case, we formulate a necessary and sufficient dual criterion of subtransversality in general Banach spaces. Our more general results suggest an intermediate notion of subtransversality, what we call weak intrinsic subtransversality, which lies between intrinsic transversality and subtransversality in Asplund spaces., Comment: 23 pages

In this paper we carry out an asymptotic analysis of the proximal-gradient dynamical system \begin{equation*}\left\{ \begin{array}{ll} \dot x(t) +x(t) = \prox_{\gamma f}\big[x(t)-\gamma\nabla\Phi(x(t))-ax(t)-by(t)\big],\\ \dot y(t)+ax(t)+by(t)=0 \end{array}\right.\end{equation*} where $f$ is a proper, convex and lower semicontinuous function, $\Phi$ a possibly nonconvex smooth function and $\gamma, a$ and $b$ are positive real numbers. We show that the generated trajectories approach the set of critical points of $f+\Phi$, here understood as zeros of its limiting subdifferential, under the premise that a regularization of this sum function satisfies the Kurdyka-\L{}ojasiewicz property. We also establish convergence rates for the trajectories, formulated in terms of the \L{}ojasiewicz exponent of the considered regularization function., Comment: arXiv admin note: substantial text overlap with arXiv:1507.01416

Locating proximal points is a component of numerous minimization algorithms. This work focuses on developing a method to find the proximal point of a convex function at a point, given an inexact oracle. Our method assumes that exact function values are at hand, but exact subgradients are either not available or not useful. We use approximate subgradients to build a model of the objective function, and prove that the method converges to the true prox-point within acceptable tolerance. The subgradient g k used at each step k is such that the distance from g k to the true subdifferential of the objective function at the current iteration point is bounded by some fixed e > 0. The algorithm includes a novel tilt-correct step applied to the approximate subgradient.

The subdifferential of a function is a generalization for nonsmooth functions of the concept of gradient. It is frequently used in variational analysis, particularly in the context of nonsmooth optimization. The present work proposes algorithms to reconstruct a polyhedral subdifferential of a function from the computation of finitely many directional derivatives. We provide upper bounds on the required number of directional derivatives when the space is $\R^1$ and $\R^2$, as well as in $\R^n$ where subdifferential is known to possess at most three vertices.

In this paper, we will study some variational properties of the Ky Fan $k$-norm $\theta=\|\cdot\|_{(k)}$ of matrices, which are closed related to a class of basic nonlinear optimization problems involving the Ky Fan $k$-norm. In particular, for the basic nonlinear optimization problems, we will introduce the concept of nondegeneracy, strict complementarity and the critical cones associated with the generalized equations. Finally, we present the explicit formulas of the conjugate function of the parabolic second order directional derivative of $\theta$, which will be referred to as the sigma term of the second order optimality conditions. The results obtain in this paper provide the necessary theoretical foundations for future work on sensitivity and stability analysis of the nonlinear optimization problems involving the Ky Fan $k$-norm., Comment: 35 pages

In this paper we present a variant of the proximal forward-backward splitting iteration for solving nonsmooth optimization problems in Hilbert spaces, when the objective function is the sum of two nondifferentiable convex functions. The proposed iteration, which will be called Proximal Subgradient Splitting Method, extends the classical subgradient iteration for important classes of problems, exploiting the additive structure of the objective function. The weak convergence of the generated sequence was established using different stepsizes and under suitable assumptions. Moreover, we analyze the complexity of the iterates., 18 pages

OA hybrid We derive stability criteria for saddle points of a class of nonsmooth optimization problems in Hilbert spaces arising in PDE-constrained optimization, using metric regularity of infinite-dimensional set-valued mappings. A main ingredient is an explicit pointwise characterization of the regular coderivative of the subdifferential of convex integral functionals. This is applied to several stability properties for parameter identification problems for an elliptic partial differential equation with non-differentiable data fitting terms.

Berge’s maximum theorem is an important statement in set-valued analysis that has significant applications in mathematical economics, operations research, and control. For a minimization problem for a continuous function of two variables and continuous compact-valued set-valued mapping defining feasible sets, this theorem states continuity of the value function and upper semi-continuity of the solution multifunction. One of the main limitations of this theorem is that the set-valued mapping is compact-valued. Recently the authors of this paper and their coauthors generalized Berge’s maximum theorem to set-valued mappings that may not be compact-valued. Here we formulate and prove the local Berge’s maximum theorem for possibly noncompact feasible sets and show that it is more general than the recently established Berge’s maximum theorem for possibly noncompact feasible sets and than the known formulations of the local Berge’s maximum theorem.