Clarke Subdifferential Research Articles

Optimality Conditions for Approximate Pareto Solutions of a Nonsmooth Vector Optimization Problem with an Infinite Number of Constraints

In this paper, we present some new necessary and sufficient optimality conditions in terms of Clarke subdifferentials for approximate Pareto solutions of a nonsmooth vector optimization problem which has an infinite number of constraints. As a consequence, we obtain optimality conditions for the particular cases of cone-constrained convex vector optimization problems and semidefinite vector optimization problems. Examples are given to illustrate the obtained results.

Strong Karush–Kuhn–Tucker Optimality Conditions for Borwein Properly Efficient Solutions of Multiobjective Semi-infinite Programming

In this paper, we study strong Karush–Kuhn–Tucker optimality conditions for Borwein properly efficient solutions of multiobjective semi-infinite programming. By using some regularity conditions in the sense of Mordukhovich subdifferentials and Clarke subdifferentials, we obtain some strong necessary optimality conditions for Borwein properly efficient solutions. Some examples are also provided to illustrate that our regularity conditions have more advantages than the previous regularity conditions in some cases.

Generalized derivatives of the optimal value of a linear program with respect to matrix coefficients

We present here a characterization of the Clarke subdifferential of the optimal value function of a linear program as a function of matrix coefficients. We generalize the result of Freund (1985) to the cases where derivatives may not be defined because of the existence of multiple primal or dual solutions.

The subdifferential of measurable composite max integrands and smoothing approximation

The subdifferential calculus for the expectation of nonsmooth random integrands involves many fundamental and challenging problems in stochastic optimization. It is known that for Clarke regular integrands, the Clarke subdifferential of the expectation equals the expectation of their Clarke subdifferential. In particular, this holds for convex integrands. However, little is known about the calculation of Clarke subgradients for the expectation of non-regular integrands. The focus of this contribution is to approximate Clarke subgradients for the expectation of random integrands by smoothing methods applied to the integrand. A framework for how to proceed along this path is developed and then applied to a class of measurable composite max integrands. This class contains non-regular integrands from stochastic complementarity problems as well as stochastic optimization problems arising in statistical learning.

Relaxation in nonconvex optimal control problems governed by evolution inclusions with the difference of two Clarke's subdifferentials

This paper considers the minimisation problem of an integral functional with an integrand and a mixed nonconvex control constraint under some suitable conditions. We minimise the integral functional on the solution set of a control system described by nonlinear evolutionary inclusions with the mixed nonconvex control constraint, in which the left-hand side of the system contains the difference of two Clarke's Subdifferentials. We discuss a relaxation problem along with the original problem and show that the relaxation problem has an optimal solution, and for every optimal solution, there exists a minimising sequence of the original problem that converges to this solution concerning the trajectory, the control, and the functional.

Linear Structure of Functions with Maximal Clarke Subdifferential

In this paper we establish that the set of Lipschitz functions $f:\mathcal{U}\rightarrow \mathbb{R}$ ($\mathcal{U}$ a nonempty open subset of $\ell_{d}^{1}$) with maximal Clarke subdifferential contains a linear subspace of uncountable dimension (in particular, an isometric copy of $\ell^{\infty}(\mathbb{N})$). This result follows along a similar line to that of a previous result of Borwein and Wang (see [Proc. Amer. Math. Soc., 128 (2000), pp. 3221--3229; Bull. Aust. Math. Soc., 72 (2005), pp. 491--496]). However, while the latter was based on Baire's category theorem, our current approach is constructive and is not linked to uniform convergence. In particular, we establish lineability (and spaceability for the Lipschitz norm) of the above set inside the set of all Lipschitz continuous functions.

On Evolution Equations Having Hypomonotonicities of Opposite Sign Governed by Sweeping Processes

We prove the local existence of solutions of a sweeping process involving a locally prox-regular set of constraints with an upper semicontinuous set-valued perturbation contained in the Clarke subdifferential of a nonconvex function. The study requires the quantified concept of local prox-regularity for the set of constraints. This paper can be considered as an improvement of previous works, since this existence (local) result is established without compactness condition on the set of constraints.

Some characterizations of robust solution sets for uncertain convex optimization problems with locally Lipschitz inequality constraints

<p style='text-indent:20px;'>In this paper, we consider an uncertain convex optimization problem with a robust convex feasible set described by locally Lipschitz constraints. Using robust optimization approach, we give some new characterizations of robust solution sets of the problem. Such characterizations are expressed in terms of convex subdifferentails, Clarke subdifferentials, and Lagrange multipliers. In order to characterize the solution set, we first introduce the so-called pseudo Lagrangian function and establish constant pseudo Lagrangian-type property for the robust solution set. We then used to derive Lagrange multiplier-based characterizations of robust solution set. By means of linear scalarization, the results are applied to derive characterizations of weakly and properly robust efficient solution sets of convex multi-objective optimization problems with data uncertainty. Some examples are given to illustrate the significance of the results.

Degenerate type fractional evolution hemivariational inequalities and optimal controls via fractional resolvent operators

ABSTRACTIn this paper, we mainly investigate controlled fractional evolution hemivariational inequalities of degenerate type in Caputo and Riemann–Liouville fractional derivatives of order in respectively. With the help of properties on fractional resolvent operators and generalised Clarke subdifferential, suitable concepts of solutions to addressed systems are formulated and existence results are established. And then, we present the existence of optimal state-control pairs for the limited Lagrange optimal systems governed by fractional evolution hemivariational inequalities of degenerate type. The optimal control results are attained by the compactness of some corresponding fractional resolvent operators.

Necessary Optimality Conditions for a Class of Nonsmooth Vector Optimization Problems

A class of nonsmooth vector optimization problems are considered, where the feasible set defined by cone constraint and the objective and constraint functions are locally Lipschitz. The concept of the Clarke's generalized directional derivative for a locally Lipschitz function is introduced. By using this concept, necessary optimality condition for the unconstrained optimization problem is established. Furthermore, a Slater-type constraint qualification is given in such a way that they generalize the classical one, when the constraint functions are differentiable. Then, Kuhn–Tucker necessary optimality condition in terms of the Clarke subdifferentials is obtained.

A New Generalized Derivative and Applications

We introduce a new generalized derivative approximated by subdifferentials of local convexifications of nonconvex functions, which overcomes "too large" Clarke subdifferential. At last we give its applications in optimization.

Nonhomogeneous Hemivariational Inequalities with Indefinite Potential and Robin Boundary Condition

We consider a nonlinear, nonhomogeneous Robin problem with an indefinite potential and a nonsmooth primitive in the reaction term. In fact, the right-hand side of the problem (reaction term) is the Clarke subdifferential of a locally Lipschitz integrand. We assume that asymptotically this term is resonant with respect the principal eigenvalue (from the left). We prove the existence of three nontrivial smooth solutions, two of constant sign and the third nodal. We also show the existence of extremal constant sign solutions. The tools come from nonsmooth critical point theory and from global optimization (direct method).

$$\ell _p$$ ℓ p Regularized low-rank approximation via iterative reweighted singular value minimization

In this paper we study the $$\ell _p$$ (or Schatten-p quasi-norm) regularized low-rank approximation problems. In particular, we introduce a class of first-order stationary points for them and show that any local minimizer of these problems must be a first-order stationary point. In addition, we derive lower bounds for the nonzero singular values of the first-order stationary points and hence also of the local minimizers of these problems. The iterative reweighted singular value minimization (IRSVM) methods are then proposed to solve these problems, whose subproblems are shown to have a closed-form solution. Compared to the analogous methods for the $$\ell _p$$ regularized vector minimization problems, the convergence analysis of these methods is significantly more challenging. We develop a novel approach to establishing the convergence of the IRSVM methods, which makes use of the expression of a specific solution of their subproblems and avoids the intricate issue of finding the explicit expression for the Clarke subdifferential of the objective of their subproblems. In particular, we show that any accumulation point of the sequence generated by the IRSVM methods is a first-order stationary point of the problems. Our computational results demonstrate that the IRSVM methods generally outperform the recently developed iterative reweighted least squares methods in terms of solution quality and/or speed.

Convergence of Rothe scheme for a class of dynamic variational inequalities involving Clarke subdifferential

ABSTRACTIn the first part of the paper we deal with a second-order evolution variational inequality involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem. Next, we show that the solution is unique and the convergence is strong. In the second part of the paper, we consider a dynamic visco-elastic problem of contact mechanics. We assume that the contact process is governed by a normal damped response condition with a unilateral constraint and the body is non-clamped. The mechanical problem in its weak formulation reduces to a variational–hemivariational inequality that can be solved by finding a solution of a corresponding abstract problem related to one studied in the first part of the paper. Hence, we apply obtained existence result to provide the weak solvability of contact problem.

Convergence of a Time Discretization for a Nonlinear Second-Order Inclusion

AbstractWe study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multi-valued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second-order-in-time differential inclusion involving a Clarke subdifferential of a locally Lipschitz, possibly non-convex and non-smooth potential. In the two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

Weak solvability of a fractional viscoelastic frictionless contact problem

The goal of this paper is to study a quasistatic frictionless contact problem for a viscoelastic body in which the constitutive equation is modeled with the fractional Kelvin–Voigt law and the contact condition is described by the Clarke subdifferential of a nonconvex and nonsmooth functional. The variational formulation of this problem is provided in the form of a fractional hemivariational inequality. In order to solve this inequality, we apply the Rothe method and prove that the associated abstract Volterra inclusion has at least one solution.

Existence of solutions for a second order boundary value problem with the Clarke subdifferential

In this paper, we prove a theorem on the existence of solutions for a second order differential inclusion governed by the Clarke subdifferential of a Lipschitzian function and by a mixed semicontinuous perturbation.

(Sub-)Gradient Formulae for Probability Functions of Random Inequality Systems under Gaussian Distribution

We consider probability functions of parameter-dependent random inequality systems under Gaussian distribution. As a main result, we provide an upper estimate for the Clarke subdifferential of such probability functions without imposing compactness conditions. A constraint qualification ensuring continuous differentiability is formulated. Explicit formulae are derived from the general result in the case of linear random inequality systems. In the case of a constant coefficient matrix, an upper estimate for even the smaller Mordukhovich subdifferential is proven.

Separation Functions and Optimality Conditions in Vector Optimization

In this paper, we propose weak separation functions in the image space for general constrained vector optimization problems on strong and weak vector minimum points. Gerstewitz function is applied to construct a special class of nonlinear separation functions as well as the corresponding generalized Lagrangian functions. By virtue of such nonlinear separation functions, we derive Lagrangian-type sufficient optimality conditions in a general context. Especially for nonconvex problems, we establish Lagrangian-type necessary optimality conditions under suitable restriction conditions, and we further deduce Karush–Kuhn–Tucker necessary conditions in terms of Clarke subdifferentials.

Solvability and optimal controls for semilinear fractional evolution hemivariational inequalities

This paper is concerned with the control systems of semilinear fractional evolution hemivariational inequalities and their optimal controls in Banach space. Firstly, the existence of mild solutions is obtained and proved mainly by using a well‐known fixed point theorem of multivalued maps and the properties of generalized Clarke subdifferential. Then, by applying generally mild conditions of cost functionals, we investigate the existence results of the optimal controls for fractional differential evolution hemivariational inequalities. Finally, an example is given to demonstrate the applicability of the main results. Copyright © 2016 John Wiley &amp; Sons, Ltd.