Nonsmoothness is often a curse for optimization; but it is sometimes a blessing, in particular for applications in machine learning. In this paper, we present the specific structure of nonsmooth optimization problems appearing in machine learning and illustrate how to leverage this structure in practice, for compression, acceleration, or dimension reduction. We pay a special attention to the presentation

In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We

In a Hilbert framework ℌ, we study the convergence properties of a Newton-like inertial dynamical system governed by a general maximally monotone operator A ℌ: → 2ℌ. When A is equal to the subdifferential of a convex lower semicontinuous proper function, the dynamic corresponds to the introduction of the Hessian-driven damping in the continuous version of the accelerated gradient method of Nesterov

We focus on a very general problem in the theory of dynamic systems, namely that of studying measure differential inclusions with varying measures. The multifunction on the right hand side has compact non-necessarily convex values in a real Euclidean space and satisfies bounded variation hypotheses with respect to the Pompeiu excess (and not to the Hausdorff-Pompeiu distance, as usually in literature)

The present paper shows how the linear independence constraint qualification (LICQ) can be combined with some conditions put on the first-order and second-order derivatives of the objective function and the constraint functions to ensure the Robinson stability and the Lipschitz-like property of the stationary point set map of a general C2-smooth parametric constrained optimization problem. So, a part

Variational Analysis studies mathematical objects under small variations. With regards to optimization, these objects are typified by representations of first-order or second-order information (gradients, subgradients, Hessians, etc). On the other hand, Derivative-Free Optimization studies algorithms for continuous optimization that do not use first-order information. As such, researchers might conclude

In this paper we study existence of solutions to the generalized equation 0 ∈ f(p,x) + F(x), where f is a function, F is a set-valued mapping, and p is a parameter. Conditions are given, in terms of metric regularity of F, local convex-valuedness of F− 1, and partial calmness of f with respect to x uniformly in p, for the property that, for any p near the reference value, the generalized equation has

Using the result of the error estimate of the simple extended Newton method established in the present paper for solving abstract inequality systems, we study the error bound property of approximate solutions of abstract inequality systems on Banach spaces with the involved function F being Fréchet differentiable and its derivative \(F^{\prime }\) satisfying the center-Lipschitz condition (not necessarily

Probability constraints are a popular modelling mechanism in applications. They help to model feasible decisions when the latter are taken prior to observing uncertainty and both decisions and uncertainty are involved in a constraint structure of an optimization problem. The popularity of this paradigm is attested by a vast literature using probability constraints. In this work we try to provide, with

We consider the problem of minimization for a function with Lipschitz continuous gradient on a proximally smooth and smooth manifold in a finite dimensional Euclidean space. We consider the Lezanski-Polyak-Lojasiewicz (LPL) conditions in this problem of constrained optimization. We prove that the gradient projection algorithm for the problem converges with a linear rate when the LPL condition holds

This paper concerns the interconnection between the positive definiteness of limiting coderivative and the local strong maximal monotonicity of set-valued mappings suspected in Mordukhovich and Nghia (SIAM J. Optim. 26, 1032–1059, 2016, Conjecture 3.6). We disprove the conjecture by a counterexample and provide some special classes at which it is true. However, the positive definiteness of limiting

We study the pointwise partial ordering of representative functions for a monotone operator and in particular we focus on the bigger conjugate representative functions that represent a fixed initial (non-maximal) monotone operator. The first problem considered is that of constructing a new representative function from a given member of this class when wanting to add an additional monotonically related

The aim of this survey is to present the main important techniques and tools from variational analysis used for first and second order dynamical systems of implicit type for solving monotone inclusions and non-smooth optimization problems. The differential equations are expressed by means of the resolvent (in case of a maximally monotone set valued operator) or the proximal operator for non-smooth

This paper studies solution stability of a parametric optimal control problem governed by semilinear elliptic equations and nonconvex objective function with mixed pointwise constrains in which the controls act both in the domain and on the boundary. We give sufficient conditions under which the solution map is upper semicontinuous and continuous in parameters.

The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Hölder case is given a special attention. Our main objective is not formally extending our earlier results from the Hölder to a more

In this paper we prove the existence of solutions of regularized set-valued variational inequalities involving Brézis pseudomonotone operators in reflexive and locally uniformly convex Banach spaces. By taking advantage of this result, we approximate a general set-valued variational inequality with suitable regularized set-valued variational inequalities, and we show that their solutions weakly converge

In this paper, we state, in separable Hilbert spaces, the existence of absolutely continuous solutions for a couple of evolution problems governed by time and state dependent maximal monotone operator and closed convex sweeping process, with perturbations.

In this paper, we propose a novel splitting method for finding a zero point of the sum of two monotone operators where one of them is Lipschizian. The weak convergence the method is proved in real Hilbert spaces. Applying the proposed method to composite monotone inclusions involving parallel sums yields a new primal-dual splitting which is different from the existing methods. Connections to existing

We consider inertial iterative schemes for approximating a fixed point of any given non-expansive operator in real Hilbert spaces. We provide new conditions on the inertial factors that ensure weak convergence and depend only on the iteration coefficients. For the special case of the Douglas-Rachford splitting, the conditions boil down to a sufficiently small upper bound on the sequence of inertial

We use the concept of barrier-based smoothing approximations to extend the non-interior continuation method, which was proposed by B. Chen and N. Xiu for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions, to an inexact non-interior continuation method for variational inequalities over general closed convex sets. Newton equations involved in the method are solved inexactly

In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of set-valued iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Hölder regularity of the minimum time function in optimal control. We then apply such result to prove Hölder continuity

The paper addresses continuous-time nonlinear programming problems with equality and inequality constraints. First and second order necessary optimality conditions are obtained under a constant rank type constraint qualification. The first order necessary conditions are of Karush-Kuhn-Tucker type.

An evolution inclusion with the right-hand side containing a time-dependent maximal monotone operator and a multivalued mapping with closed nonconvex values is studied in a separable Hilbert space. The dependence of the maximal monotone operator on time is described with the help of the distance between maximal monotone operators in the sense of Vladimirov. This distance as a function of time has bounded

We introduce and develop a notion of “catastrophic radii” to identify where a minimization method may require an arbitrarily large number of steps to approximate a minimizer of an objective function, and we use this notion to categorize the performance of method/objective combinations. In order to investigate the different categories, we explore simple examples where explicit formulas can be used,

Influenced by a recent note by M. Ivanov and N. Zlateva, we prove a statement in the style of Nash-Moser-Ekeland theorem for mappings from a Fréchet-Montel space with values in any Fréchet space (not necessarily standard). The mapping under consideration is supposed to be continuous and directionally differentiable (in particular Gateaux differentiable) with the derivative having a right inverse. We

Motivated by structures that appear in deep neural networks, we investigate nonlinear composite models alternating proximity and affine operators defined on different spaces. We first show that a wide range of activation operators used in neural networks are actually proximity operators. We then establish conditions for the averagedness of the proposed composite constructs and investigate their asymptotic

Motivated by a number of questions concerning transversality-type properties of pairs of sets recently raised by Ioffe and Kruger, this paper reports several new characterizations of the intrinsic transversality property in Hilbert spaces. New results in terms of normal vectors clarify the picture of intrinsic transversality, its variants and sufficient conditions for subtransversality, and unify several

One of the purposes in this paper is to provide a better understanding of the alternance property which occurs in Chebyshev polynomial approximation and continuous piecewise polynomial approximation problems. In the first part of this paper, we prove that alternating sequences of any continuous function are finite in any given segment and then propose an original approach to obtain new proofs of the

The notion of set-valued means is introduced. Set-valued counterparts of the arithmetic, quasi-arithmetic and Lagrangian means are investigated and various properties of them are presented.

In this paper, we study the existence of solutions for evolution inclusions governed by time-dependent maximal monotone operators with a full domain. Without assumptions concerning time-regularity on the time-dependent maximal monotone operators, and by using the Moreau-Yosida regularization technique, we establish the existence of solutions in Hilbert spaces. The theoretical result is applied to prove

Epi/hypo-convergence is extended to the case of bifunctions defined on general domains. Its basic characterizations are established. Variational properties such as those about saddle points, weak saddle points, minsup-points, sup-projections, etc, of bifunctions are shown to be preserved for their epi/hypo-limits (possibly under some additional assumptions). Approximations of quasi-equilibrium problems

We consider problems of linear copositive programming where feasible sets consist of vectors for which the quadratic forms induced by the corresponding linear matrix combinations are nonnegative over the nonnegative orthant. Given a linear copositive problem, we define immobile indices of its constraints and a normalized immobile index set. We prove that the normalized immobile index set is either

For any linear transformation and two convex closed sets, we provide necessary and sufficient conditions for the transformation of the intersection of the sets to coincide with the intersection of their images. We also identify conditions for non-convex closed sets, continuous transformations, and multiple sets. We demonstrate the usefulness of our results via an application to the economics literature

Using a new implicit discretization scheme, we study in this paper the existence and uniqueness of strong solutions for a class of Lur’e dynamical systems where the set-valued feedback depends on both the time and the state. This work is a generalization of Tanwani et al. (SIAM J. Control Opti. 56(2), 751–781, 2018) where the time-dependent set-valued feedback is considered to acquire weak solutions

This paper investigates a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the literature. Some slope and abstract coderivative characterizations are established.

We take a fresh look at the Bartle-Graves theorem pointing out the main differences with the standard implicit function theorem. We then present a set-valued version of this theorem which generalizes some recent results. Applications to variational inequalities and differential inclusions are also given.

There is a basic paradigm, called here the radius of well-posedness, which quantifies the “distance” from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we

This article is devoted to the analysis of necessary and/or sufficient conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova quasidifferentials. We obtain new necessary and sufficient conditions for the local metric regularity of a multifunction in terms of quasidifferentials of the distance function to this multifunction. We also propose a new MFCQ-type constraint qualification for

In this paper, we study the mathematical program with equilibrium constraints 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

When restricted to a subspace, a nonsmooth function can be differentiable. It is known that for a nonsmooth convex function and a point, the Euclidean space can be decomposed into two subspaces: \(\mathcal {U}\), over which a special Lagrangian can be defined and has nice smooth properties and \(\mathcal {V}\), the orthogonal complement subspace of \(\mathcal {U}\). In this paper we generalize the

We introduce a projection-type algorithm for solving the variational inequality problem for point-to-set operators, and establish its convergence properties. Namely, we assume that the operator of the variational inequality is continuous in the point-to-set sense, i.e., inner- and outer-semicontinuous. Under the assumption that the dual solution set is not empty, we prove that our method converges

In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are

In this paper we associate with an infinite family of real extended functions defined on a locally convex space a sum, called robust sum, which is always well-defined. We also associate with that family of functions a dual pair of problems formed by the unconstrained minimization of its robust sum and the so-called optimistic dual. For such a dual pair, we characterize weak duality, zero duality gap

We give new criteria for weak and strong invariant closed sets for differential inclusions in \(\mathbb {R}^{n}\), and which are simultaneously governed by Lipschitz Cusco mapping and by maximal monotone operators. Correspondingly, we provide different characterizations for the associated strong Lyapunov functions and pairs. The resulting conditions only depend on the data of the system, while the

This article proposes a new algorithm for computing solutions to mixed flow problems involving the classical Lighthill-Whitham-Richards (LWR) flow model. We show that the behavior of autonomous vehicles can be described by viability constrained solutions to an alternate Hamilton-Jacobi formulation of the same model, for appropriate upper constraint functions. We also provide the physical interpretation

This article presents a survey of several properties of the set of solutions for a differential inclusion involving a time-delayed component and with right-hand side parametrized by either an upper semicontinuous or lower semicontinuous multifunction. Our results include: existence of solutions, compactness and contractibility of the solution and reachable sets in the upper semicontinuous case, precompactness

For solving monotone inclusion problems, we propose an inertial under-relaxed version of the relative-error hybrid proximal extragradient method. We study the asymptotic convergence of the method, as well as its nonasymptotic global convergence rates in terms of iteration complexity. We analyze the new method under more flexible assumptions than existing ones, both on the extrapolation and on the relative-error

We investigate the space of convex minimal usco maps from a Tychonoff space to the space of real numbers. Its elements are set-valued maps that are important e.g. in the study of subdifferentials of convex functions. We show that if the underlying space is normal, convex minimal usco maps can be approximated in the Vietoris topology by continuous functions. Using the strong Choquet game we prove complete

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

In this paper, strongly pseudomonotone quasi-variational inequalities are investigated. We provide sufficient conditions for existence and uniqueness of solutions of strongly pseudomonotone quasi-variational inequalities. We present some error bounds in terms of residual and regularized gap functions. The global exponential stability of equilibrium solutions of a projected dynamical system for strongly

In this paper we select two tools of investigation of the classical metric regularity of set-valued mappings, namely the Ioffe criterion and the Ekeland Variational Principle, which we adapt to the study of the directional setting. In this way, we obtain in a unitary manner new necessary and/or sufficient conditions for directional metric regularity. As an application, we establish stability of this

The main concern of this paper is to investigate sensitivity properties of parametric evolution systems of first order involving a general class of nonconvex functions. Using recent results on the stability of the subdifferentials, with respect to the Gamma convergence, of the associated sequence of subsmooth or semiconvex functions, we give some continuity properties of the solution set associated

Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Hölder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Hölder calmness of the argmin mapping in terms of the level set mapping (with respect

In this paper, we study bornological generalized differential properties of sets with nonsmooth boundaries, nonsmooth functions, and set-valued mappings in smooth Banach spaces. We establish a fuzzy intersection rule for bornological normal cones and develop fuzzy calculus for bornological generalized differential constructions as well as exact calculus for the limiting counterparts of these constructions

We present existence of solution and necessary conditions for an optimal control problem with a particular case of sweeping processes with constant sweeping set. Using a sequence of approximating problems we prove existence of solution and necessary conditions. The special feature of our approach resides on the nature of the approximating problems: in contrast with other literature these problems do

This paper is devoted to a generalized evolution system called fractional partial differential variational inequality which consists of a mixed quasi-variational inequality combined with a fractional partial differential equation in a Banach space. Invoking the pseudomonotonicity of multivalued operators and a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, first, we prove that the solution

We consider non-autonomous calculus of variations problems with a state constraint represented by a given closed set. We prove that if the interior of the Clarke tangent cone of the state constraint set is non-empty (this is the constraint qualification that we suggest here), then the necessary optimality conditions apply in the normal form. We establish normality results for (weak) local minimizers

Algorithms for problem decomposition and splitting in optimization and the solving of variational inequalities have largely depended on assumptions of convexity or monotonicity. Here, a way of “eliciting” convexity or monotonicity is developed which can get around that limitation. It supports a procedure called the progressive decoupling algorithm, which is derived from the proximal point algorithm

We study the invertibility nonsmooth maps between infinite-dimensional Banach spaces. To this end, we introduce an analogue of the notion of pseudo-Jacobian matrix of Jeyakumar and Luc in this infinite-dimensional setting. Using this, we obtain several inversion results. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context.

We consider nonsmooth optimization problems whose objective function is defined by the Difference of Convex (DC) functions. With the aim of computing critical points that are also d(irectional)-stationary for such a class of nonconvex programs we propose an algorithmic scheme equipped with an inertial-force procedure. In contrast to the classical DC algorithm of P. D. Tao and L. T. H. An, the proposed