Akhtar A. Khan

GENERALIZED CONTINGENT EPIDERIVATIVES IN SET-VALUED OPTIMIZATION: OPTIMALITY CONDITIONS

ABSTRACT Necessary and sufficient optimality conditions are given for various optimality notions in set-valued optimization. These optimality conditions are given by employing the generalized contingent epiderivative and the weak contingent epiderivative of the objective set-valued map and the set-valued map defining the constraints. The known Lagrange multiplier rule and the so-called Zowe-Kurcyusz-Robinson (cf. Robinson, S.M. Stability Theory for Systems of Inequalities. II, Differentiable Nonlinear Systems. SIAM J. Numer. Anal. 1976, 13, 497–513. Zowe, J.; Kurcyusz, S. Regularity and Stability for the Mathematical Programming Problem in Banach spaces, Appl. Math. Optim 1979, 5, 49–62.) regularity condition are extended using these differentiability notions.

Set-valued Optimization

Introduction.- Order Relations and Ordering Cones.- Continuity and Differentiability.- Tangent Cones and Tangent Sets.- Nonconvex Separation Theorems.- Hahn-Banach Type Theorems.- Hahn-Banach Type Theorems.- Conjugates and Subdifferentials.- Duality.- Existence Results for Minimal Points.- Ekeland Variational Principle.- Derivatives and Epiderivatives of Set-valued Maps.- Optimality Conditions in Set-valued Optimization.- Sensitivity Analysis in Set-valued Optimization and Vector Variational Inequalities.- Numerical Methods for Solving Set-valued Optimization Problems.- Applications.

An Abstract Framework for Elliptic Inverse Problems: Part 1. An Output Least-Squares Approach

The solution of an elliptic boundary value problem is an infinitely differentiable function of the coefficient in the partial differential equation. When the (coefficient-dependent) energy norm is used, the result is a smooth, convex output least-squares functional. Using total variation regularization, it is possible to estimate discontinuous coefficients from interior measurements. The minimization problem is guaranteed to have a solution, which can be obtained in the limit from finite-dimensional discretizations of the problem. These properties hold in an abstract framework that encompasses several interesting problems: the standard (scalar) elliptic BVP in divergence form, the system of isotropic elasticity, and others.

An Abstract Framework for Elliptic Inverse Problems: Part 2. An Augmented Lagrangian Approach

The coefficient in a linear elliptic partial differential equation can be estimated from interior measurements of the solution. Posing the estimation problem as a constrained optimization problem with the PDE as the constraint allows the use of the augmented Lagrangian method, which is guaranteed to converge. Moreover, the convergence analysis encompasses discretization by finite element methods, so the proposed algorithm can be implemented and will produce a solution to the constrained minimization problem. All of these properties hold in an abstract framework that encompasses several interesting problems: the standard (scalar) elliptic BVP in divergence form, the system of isotropic elasticity, and others. Moreover, the analysis allows for the use of total variation regularization, so rapidly-varying or even discontinuous coefficients can be estimated.

A New Convex Inversion Framework for Parameter Identification in Saddle Point Problems with an Application to the Elasticity Imaging Inverse Problem of Predicting Tumor Location

This work presents a thorough theoretical and numerical analysis of the elasticity imaging inverse problem of tumor identification in the soft tissue of the human body. Beyond the obvious merits of its applications, this problem also presents significant mathematical challenges. The near incompressibility inherent in the model of linear elasticity in the body gives rise to the “locking effect” and necessitates a unique treatment of both the direct and inverse problems. A general optimization framework for the identification of parameters in saddle point problems is presented along with a new modified output least-squares (MOLS) objective functional. The MOLS functional is shown to be convex, thus overcoming the nonconvexity of the classical output least-squares (OLS) functional, and the new framework is shown to be capable of accommodating both smooth and discontinuous parameters. Generalized derivative formulas for the coefficient-to-solution map are also given along with a complete convergence analysis....

The existence of contingent epiderivatives for set-valued maps☆

This short note deals with the issue of existence of contingent epiderivatives for set-valued maps defined from a real normed space to the real line. A theorem of Jahn-Rauh [1], given for the existence of contingent epiderivatives, is used to obtain more general existence results. The strength and the limitations of the main result are discussed by means of some examples.

SOME CALCULUS RULES FOR CONTINGENT EPIDERIVATIVES

This article presents some calculus rules for contingent epiderivatives of set-valued maps. Among other results the main emphasis is focused on a formula for scalar multiplication, sum formulae and chain rules. The calculus of contingent cones and some inversion theorems are used as a tool. Some applications are also given.

An equation error approach for the elasticity imaging inverse problem for predicting tumor location

The primary objective of this work is to study the elasticity imaging inverse problem of identifying cancerous tumors in the human body. This nonlinear inverse problem not only represents an important and interesting application, it also brings forth noteworthy mathematical challenges since the underlying model is a system of elasticity equations involving incompressibility. Due to the locking effect, classical finite element methods are not effective for incompressible elasticity equations. Therefore, special treatment is necessary for both the direct and inverse problems. To study the inverse problem in an optimization framework, we propose an extension of the equation error approach. We focus on two cases, namely when the material parameter is sufficiently smooth and when it is may be discontinuous. For the latter case, we extend the total variation regularization method to the elasticity imaging inverse problem. We give the existence results for the proposed equation error approach and give the convergence analysis for the discretized problem. We give sufficient details on the discrete formulas as well as on the implementation issues. Numerical examples for smooth and discontinuous coefficients are given.

Dubovitskii-Milyutin Approach in Set-Valued Optimization

By exploring the ideas around the Dubovitskii-Milyutin approach, necessary optimality conditions are given for various optimality notions in set-valued optimization. These optimality conditions are given by using the contingent derivative and the generalized contingent epiderivative of the objective set-valued map and the set-valued maps defining the constraints. The notions of subgradients and scalarized subgradients for set-valued maps are proposed and used to state some regularity conditions.

Local minimizers versus X -local minimizers

An important property known, among other cases, for W1,p(Ω) versus