Morrey Regularity of Solutions of Nonlinear Elliptic Systems of Arbitrary Order with Restrictions on the Modulus of Ellipticity

We investigate the dependence of the regularity of generalized solutions of nonlinear elliptic systems on the modulus of ellipticity and regularity of the right-hand side. We establish Morrey regularity with limit exponent determined by the modulus of ellipticity in the case where the right-hand side belongs to a space with a norm stronger than the Dini function. These conditions are exact for second-order systems, namely, for any violation of the Dini condition, we construct a solution that does not belong to the Morrey space with limit exponent.     

On necessary extremum conditions for finite-dimensional problems with inequality constraints

The finite-dimensional optimization problem with equality and inequality constraints is examined. The case where the classical regularity condition is violated is analyzed. Necessary second-order extremum conditions are obtained that are stronger versions of some available results.     

A Transformation Approach in Shape Optimization: Existence and Regularity Results

In this article, we consider a model shape optimization problem. The state variable solves an elliptic equation on a star-shaped domain, where the radius is given via a control function. First, we reformulate the problem on a fixed reference domain, where we focus on the regularity needed to ensure the existence of an optimal solution. Second, we introduce the Lagrangian and use it to show that the optimal solution possesses a higher regularity, which allows for the explicit computation of the derivative of the reduced cost functional as a boundary integral. We finish the article with some second-order optimality conditions.     

On orthogonal polynomials with respect to the form -(x - c)S'

Summary A form (linear functional) $u$ is called regular if we can associate with it a sequence of monic orthogonal polynomials. On certain regularity conditions, the product of a non regular form by a polynomial can be regular. The purpose of this work is to establish regularity conditions of the form $-(x-c){\mathbf S}',$ where ${\mathbf S}$ is a classical (Bessel, Jacobi). We give the second-order recurrence relations and structure relations of its corresponding orthogonal polynomial sequence. We conclude with an example as an illustration.     

On strong and weak second-order necessary optimality conditions for nonlinear programming

Second-order necessary optimality conditions play an important role in optimization theory. This is explained by the fact that most numerical optimization algorithms reduce to finding stationary points satisfying first-order necessary optimality conditions. As a rule, optimization problems, especially the high dimensional ones, have a lot of stationary points so one has to use second-order necessary optimality conditions to exclude nonoptimal points. These conditions are closely related to second-order constraint qualifications, which guarantee the validity of second-order necessary optimality conditions. In this paper, strong and weak second-order necessary optimality conditions are considered and their validity proved under so-called critical regularity condition at local minimizers.     

Second-order conditions for nonsmooth multiobjective optimization problems with inclusion constraints

This paper investigates second-order optimality conditions for general multiobjective optimization problems with constraint set-valued mappings and an arbitrary constraint set in Banach spaces. Without differentiability nor convexity on the data and with a metric regularity assumption the second-order necessary conditions for weakly efficient solutions are given in the primal form. Under some additional assumptions and with the help of Robinson -Ursescu open mapping theorem we obtain dual second-order necessary optimality conditions in terms of Lagrange-Kuhn-Tucker multipliers. Also, the second-order sufficient conditions are established whenever the decision space is finite dimensional. To this aim, we use the second-order projective derivatives associated to the second-order projective tangent sets to the graphs introduced by Penot. From the results obtained in this paper, we deduce and extend, in the special case some known results in scalar optimization and improve substantially the few results known in vector case.     

Optimality conditions for nonconvex semidefinite programming

This paper concerns nonlinear semidefinite programming problems for which no convexity assumptions can be made. We derive first- and second-order optimality conditions analogous to those for nonlinear programming. Using techniques similar to those used in nonlinear programming, we extend existing theory to cover situations where the constraint matrix is structurally sparse. The discussion covers the case when strict complementarity does not hold. The regularity conditions used are consistent with those of nonlinear programming in the sense that the conventional optimality conditions for nonlinear programming are obtained when the constraint matrix is diagonal. Received: May 15, 1998 / Accepted: April 12, 2000?Published online May 12, 2000     

Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization

In this paper, we propose the concept of a second-order composed contingent derivative for set-valued maps, discuss its relationship to the second-order contingent derivative and investigate some of its special properties. By virtue of the second-order composed contingent derivative, we extend the well-known Lagrange multiplier rule and the Kurcyusz–Robinson–Zowe regularity assumption to a constrained set-valued optimization problem in the second-order case. Simultaneously, we also establish some second-order Karush–Kuhn–Tucker necessary and sufficient optimality conditions for a set-valued optimization problem, whose feasible set is determined by a set-valued map, under a generalized second-order Kurcyusz–Robinson–Zowe regularity assumption.     

Metric regularity,tangent sets,and second-order optimality conditions

A strong regularity theorem is proved, which shows that the usual constraint qualification conditions ensuring the regularity of the set-valued maps expressing feasibility in optimization problems, are in fact minimal assumptions. These results are then used to derive calculus rules for second-order tangent sets, allowing us in turn to obtain a second-order (Lagrangian) necessary condition for optimality which completes the usual one of positive semidefiniteness on the Hessian of the Lagrangian function.On leave from Universidad de Chile, Casilla 170/3 Correo 3, Santiago, Chile.     

Differential Conditions for Constrained Nonlinear Programming via Pareto Optimization

We deal with the differential conditions for local optimality. The conditions that we derive for inequality constrained problems do not require constraint qualifications and are the broadest conditions based on only first-order and second-order derivatives. A similar result is proved for equality constrained problems, although the necessary conditions require the regularity of the equality constraints.     

Necessary conditions for an extremum in a mathematical programming problem

For minimization problems with equality and inequality constraints, first-and second-order necessary conditions for a local extremum are presented. These conditions apply when the constraints do not satisfy the traditional regularity assumptions. The approach is based on the concept of 2-regularity; it unites and generalizes the authors’ previous studies based on this concept.     

Estimate for the second-order derivatives of solutions to boundary-value problems of the theory of non-Newtonian liquids

The boundary regularity of solutions to some boundary-value problems describing stationary flow of generalized Newtonian liquids is studied. The dissipative potential is of quadratic growth at infinity. We prove that the second-order derivatives of the solution are pth power summable functions, where p is greater than two. The partial regularity of the strain velocity tensor is established. In the two-dimensional case, the complete regularity of the strain velocity tensor is also proved. Bibliography: 14 titles. Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 222–246.     

Generalized second-order directional derivatives and optimization with C1,1 functions

In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C¹,¹ functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gãteaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C^1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems.     

Unconditional bases related to a nonclassical second-order differential operator

In the present paper, we introduce the notion of regularity of boundary conditions for a simplest second-order differential equation with a deviating argument. We prove the Riesz basis property for a system of root vectors of the corresponding generalized spectral problem with regular boundary conditions (in the sense of the introduced definition). Examples of irregular boundary conditions to which the theory of Il’in basis property can be applied are given.     

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

An optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.     

Feasible Perturbations of Control Systems with Pure State Constraints and Applications to Second-Order Optimality Conditions

We propose second-order necessary optimality conditions for optimal control problems with very general state and control constraints which hold true under weak regularity assumptions on the data. In particular the pure state constraints are general closed sets, the optimal control is supposed to be merely measurable and the dynamics may be discontinuous in the time variable as well. These results are obtained by an approach based on local perturbations of the reference process by second-order tangent directions. This method allows direct and quite simple proofs.     

Second-order optimality conditions for inequality constrained problems with locally Lipschitz data

In this paper we obtain second-order optimality conditions of Fritz John and Karush–Kuhn–Tucker types for the problem with inequality constraints in nonsmooth settings using a new second-order directional derivative of Hadamard type. We derive necessary and sufficient conditions for a point [`(x)]{\bar x} to be a local minimizer and an isolated local one of order two. In the primal necessary conditions we suppose that all functions are locally Lipschitz, but in all other conditions the data are locally Lipschitz, regular in the sense of Clarke, Gateaux differentiable at [`(x)]{\bar x}, and the constraint functions are second-order Hadamard differentiable at [`(x)]{\bar x} in every direction. It is shown by an example that regularity and Gateaux differentiability cannot be removed from the sufficient conditions.     

Second-order variational analysis and characterizations of tilt-stable optimal solutions in infinite-dimensional spaces

The paper is devoted to developing second-order tools of variational analysis and their applications to characterizing tilt-stable local minimizers of constrained optimization problems infinite-dimensional spaces with many results new also in finite-dimensional settings. The importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization. Based on second-order generalized differentiation, we obtain qualitative and quantitative characterizations of tilt stability in general frameworks of constrained optimization and establish its relationships with strong metric regularity of subgradient mappings and uniform second-order growth. The results obtained are applied to deriving new necessary and sufficient conditions for tilt-stable minimizers in problems of nonlinear programming with twice continuously differentiable data in Hilbert spaces.     

Loss of regularity for p-evolution type models

The goal of the paper is to study the loss of regularity for special p-evolution type models with bounded coefficients in the principal part. The obtained loss of regularity is related in an optimal way to some unboundedness conditions for the derivatives of coefficients up to the second-order with respect to t.     

Defect correction and Galerkin's method for second-order boundary value problems

The defect correction technique, based on the Galerkin finite element method, is analyzed as a procedure to obtain highly accurate numerical solutions to second-order elliptic boundary value problems. The basic solutions, defined over a rectangular region Ω, are computed using continuous piecewise bilinear polynomials on rectangles. These solutions are O(h²) accurate globally in the second-order discrete Sobolev norm. Corrections to these basic solutions are obtained using higher-order piecewise polynomials (Lagrange polynomials or splines) to form defects. An O(h²) improvement is gained on the first correction. The lack of regularity of the discrete problems (beyond the second-order Sobolev norm) makes it impossible to retain this order of improvement, but for problems satisfying certain periodicity conditions, straightforward arbitrary accuracy is obtained, since these problems possess high-order regularity. © 1992 John Wiley & Sons, Inc.