Douglas Ward | Miami University (original) (raw)

Papers by Douglas Ward

Research paper thumbnail of Upper Bounds on a Parabolic Second Order Directional Derivative of the Marginal Function

Abstract We establish an upper bound on the parabolic second-order upper Dini derivative of the m... more Abstract We establish an upper bound on the parabolic second-order upper Dini derivative of the marginal function of a parametric nonlinear program with smooth equality constraint functions but possibly nonsmooth objective and inequality constraint functions. The main ...

Research paper thumbnail of General constraint qualifications in nondifferentiable programming

Mathematical Programming, May 1, 1990

We show that a familiar constraint qualification of differentiable programming has “nonsmooth” co... more We show that a familiar constraint qualification of differentiable programming has “nonsmooth” counterparts. As a result, necessary optimality conditions of Kuhn—Tucker type can be established for inequality-constrained mathematical programs involving functions not assumed to be differentiable, convex, or locally Lipschitzian. These optimality conditions reduce to the usual Karush—Kuhn—Tucker conditions in the differentiable case and sharpen previous results in the locally Lipschitzian case.

Research paper thumbnail of A Comparison of Second-Order Epiderivatives: Calculus and Optimality Conditions

Journal of Mathematical Analysis and Applications, Jul 1, 1995

Research paper thumbnail of Sufficient Conditions for Weak Sharp Minima of Order Two and Directional Derivatives of the Value Function

CRC Press eBooks, Sep 23, 2020

Research paper thumbnail of Which subgradients have sum formulas?

Nonlinear Analysis-theory Methods & Applications, Nov 1, 1988

Research paper thumbnail of Upper DSL approximates and nonsmooth optimization

Optimization, 1990

ABSTRACT For a tangent cone A, an extended-real-valued function f is said to admit an “A upper DS... more ABSTRACT For a tangent cone A, an extended-real-valued function f is said to admit an “A upper DSL approximate” at x if its “A directional derivative” at x is majorized by a difference of lower semicontinuous sublinear functions. By means of such approximates we establish necessary optimality conditions of Fritz John and Kuhn-Tucker type for a nonsmooth, inequality-constrained mathematical program. Optimality conditions involving the quasidifferentials of Demyanov, the upper convex approximates of Pshenichnyi, and the upper DSL approximates of A, Shapiro are among the special cases of these general optimality conditions.

Research paper thumbnail of Second-order necessary conditions in nonsmooth programming

Systems modelling and optimization, 2022

Research paper thumbnail of Convex Directional Derivatives in Optimization

Broadly speaking, a generalized convex function is one which has some property of convex function... more Broadly speaking, a generalized convex function is one which has some property of convex functions that is essential in a particular application. Two such properties are convexity of lower level sets (in the case of quasiconvex functions) and convexity of the ordinary directional derivative as a function of direction (in the case of Pshenichnyi’s quasidifferentiable functions). In recent years, several directional derivatives have been defined that, remarkably, are always convex as a function of direction.

Research paper thumbnail of Chain rules for nonsmooth functions

Journal of Mathematical Analysis and Applications, 1991

A systematic method is presented for the derivation of chain rules for compositions of functions ... more A systematic method is presented for the derivation of chain rules for compositions of functions Fof, where F is nondecreasing. This method is valid for directional derivatives and subgradients associated with any tangent cone having a short list of properties. Some major special cases are examined in detail; in particular, calculus rules are derived for Rockafellar's epi-derivatives and Clarke generalized gradients.

Research paper thumbnail of CONVEX KERNELS AND TANGENT CONE CHAIN RULES

Communications in Optimization Theory, 2023

In this paper we present formulas for the contingent and adjacent cones to the graph of a composi... more In this paper we present formulas for the contingent and adjacent cones to the graph of a composition of set-valued mappings, under conditions involving these cones' convex kernels. Special cases of the formulas include chain rules for epiderivatives of compositions of nonsmooth functions.

Research paper thumbnail of On Relations Between Vector Optimization Problems and Vector Variational Inequalities

Journal of Optimization Theory and Applications

ABSTRACT We obtain equivalences between weak Pareto solutions of vector optimization problems and... more ABSTRACT We obtain equivalences between weak Pareto solutions of vector optimization problems and solutions of vector variational inequalities involving generalized directional derivatives.

Research paper thumbnail of Weak sharp minima: characterizations and su cient conditions

Siam Journal on Control and Optimization, 1999

Research paper thumbnail of An Epigraph-Based Approach to Sensitivity Analysis in Set-Valued Optimization

Springer Proceedings in Mathematics & Statistics, 2013

ABSTRACT In this paper, we obtain estimates for the contingent and adjacent derivatives of the ep... more ABSTRACT In this paper, we obtain estimates for the contingent and adjacent derivatives of the epigraph of the marginal multifunction in parametric set-valued optimization. These estimates generalize some sensitivity results from scalar-valued optimization and provide new information in the setting of multiobjective nonlinear programming.

Research paper thumbnail of Optimization with data perturbations II. The papers are a tribute to Prof. Anthony V. Fiacco on the occasion of his 70th birthday

Annals of Operations Research

The articles of this volume will be reviewed individually. The editors present a short biography ... more The articles of this volume will be reviewed individually. The editors present a short biography and a list of selected publications of A. V. Fiacco. For Vol. I. see ibid. 27 (1990).

Research paper thumbnail of Isotone tangent cones and nonsmooth optimization

Optimization, 1987

A simple but quite general method of formulating necessary optimality conditions for an abstract ... more A simple but quite general method of formulating necessary optimality conditions for an abstract mathematical program is presented. This method centers around tangent cones which are isotone with respect to inclusion. The optimality conditions derived by this ...

Research paper thumbnail of Upper Subderivatives and Generalized Gradients of the Marginal Function of a Non-Lipschitzian Program

Annals of Operations Research - Annals OR, 2001

We obtain an upper bound for the upper subderivative of the marginal function of an abstract para... more We obtain an upper bound for the upper subderivative of the marginal function of an abstract parametric optimization problem when the objective function is lower semicontinuous. Moreover, we apply the result to a nonlinear program with right-hand side perturbations. As a result, we obtain an upper bound for the upper subderivative of the marginal function of a nonlinear program with right-hand side perturbations, which is expressed in “dual form” in terms of appropriate Lagrange multipliers. Finally, we present conditions which imply that the marginal function is locally Lipschitzian.

Research paper thumbnail of Corrigendum to "Convex Subcones of the Contingent Cone in Nonsmooth Calculus and Optimization

Transactions of the American Mathematical Society, 1989

Research paper thumbnail of Dini Derivatives of the Marginal Function of a Non-Lipschitzian Program

SIAM Journal on Optimization, 1996

ABSTRACT Upper and lower bounds are establised for the Dini directional derivatives of the margin... more ABSTRACT Upper and lower bounds are establised for the Dini directional derivatives of the marginal function of a parametric mathematical program. In this program, the equality constraint functions are assumed to be strictly differentiable, but the objective and inequality constraint functions can belong to a large class of non-Lipschitzian functions. A nonsmooth version of the Mangasarian-Fromovitz constraint qualification is also assumed. The main tool in the proofs of these bounds is the calculus of tangent cones.

Research paper thumbnail of Nonsmooth Calculus in Finite Dimensions

SIAM Journal on Control and Optimization, 1987

The notion of subgradient, originally defined for convex functions, has in recent years been exte... more The notion of subgradient, originally defined for convex functions, has in recent years been extended, via the "upper subderivative," to cover functions that are not necessarily convex or even continuous. A number of calculus rules have been proven for these generalized subgradients. This paper develops the finite-dimensional generalized subdifferential calculus for (strictly) lower semicontinuous functions under considerably weaker hypotheses than those previously used. The most general finite-dimensional convex subdifferential calculus results are recovered as corollaries. Other corollaries given include new necessary conditions for optimality in a nonsmooth mathematical program. Various chain rule formulations are considered. Equality in the subdifferential calculus formulae is proven underhypotheses weaker than the usual "subdifferential regularity" assumptions.

Research paper thumbnail of A constraint qualification in quasidifferentiable programming

Optimization, 1991

ABSTRACT One goal in quasid if Terentiable optimization is the development of optimality conditio... more ABSTRACT One goal in quasid if Terentiable optimization is the development of optimality conditions whose hypotheses are independent of the particular choice of quasidifferentials. One such hypothesis, introduced by Demyanov and Rubinov, involves the concept of a pair of convex sets being “in a general position”. In this paper, a simple condition that implies the general position hypothesis is presented. This condition is also shown to be a constraint qualification for non-asymptotic Kuhn-Tucker conditions for a quasidifferentiable program.

Research paper thumbnail of Upper Bounds on a Parabolic Second Order Directional Derivative of the Marginal Function

Abstract We establish an upper bound on the parabolic second-order upper Dini derivative of the m... more Abstract We establish an upper bound on the parabolic second-order upper Dini derivative of the marginal function of a parametric nonlinear program with smooth equality constraint functions but possibly nonsmooth objective and inequality constraint functions. The main ...

Research paper thumbnail of General constraint qualifications in nondifferentiable programming

Mathematical Programming, May 1, 1990

We show that a familiar constraint qualification of differentiable programming has “nonsmooth” co... more We show that a familiar constraint qualification of differentiable programming has “nonsmooth” counterparts. As a result, necessary optimality conditions of Kuhn—Tucker type can be established for inequality-constrained mathematical programs involving functions not assumed to be differentiable, convex, or locally Lipschitzian. These optimality conditions reduce to the usual Karush—Kuhn—Tucker conditions in the differentiable case and sharpen previous results in the locally Lipschitzian case.

Research paper thumbnail of A Comparison of Second-Order Epiderivatives: Calculus and Optimality Conditions

Journal of Mathematical Analysis and Applications, Jul 1, 1995

Research paper thumbnail of Sufficient Conditions for Weak Sharp Minima of Order Two and Directional Derivatives of the Value Function

CRC Press eBooks, Sep 23, 2020

Research paper thumbnail of Which subgradients have sum formulas?

Nonlinear Analysis-theory Methods & Applications, Nov 1, 1988

Research paper thumbnail of Upper DSL approximates and nonsmooth optimization

Optimization, 1990

ABSTRACT For a tangent cone A, an extended-real-valued function f is said to admit an “A upper DS... more ABSTRACT For a tangent cone A, an extended-real-valued function f is said to admit an “A upper DSL approximate” at x if its “A directional derivative” at x is majorized by a difference of lower semicontinuous sublinear functions. By means of such approximates we establish necessary optimality conditions of Fritz John and Kuhn-Tucker type for a nonsmooth, inequality-constrained mathematical program. Optimality conditions involving the quasidifferentials of Demyanov, the upper convex approximates of Pshenichnyi, and the upper DSL approximates of A, Shapiro are among the special cases of these general optimality conditions.

Research paper thumbnail of Second-order necessary conditions in nonsmooth programming

Systems modelling and optimization, 2022

Research paper thumbnail of Convex Directional Derivatives in Optimization

Broadly speaking, a generalized convex function is one which has some property of convex function... more Broadly speaking, a generalized convex function is one which has some property of convex functions that is essential in a particular application. Two such properties are convexity of lower level sets (in the case of quasiconvex functions) and convexity of the ordinary directional derivative as a function of direction (in the case of Pshenichnyi’s quasidifferentiable functions). In recent years, several directional derivatives have been defined that, remarkably, are always convex as a function of direction.

Research paper thumbnail of Chain rules for nonsmooth functions

Journal of Mathematical Analysis and Applications, 1991

A systematic method is presented for the derivation of chain rules for compositions of functions ... more A systematic method is presented for the derivation of chain rules for compositions of functions Fof, where F is nondecreasing. This method is valid for directional derivatives and subgradients associated with any tangent cone having a short list of properties. Some major special cases are examined in detail; in particular, calculus rules are derived for Rockafellar's epi-derivatives and Clarke generalized gradients.

Research paper thumbnail of CONVEX KERNELS AND TANGENT CONE CHAIN RULES

Communications in Optimization Theory, 2023

In this paper we present formulas for the contingent and adjacent cones to the graph of a composi... more In this paper we present formulas for the contingent and adjacent cones to the graph of a composition of set-valued mappings, under conditions involving these cones' convex kernels. Special cases of the formulas include chain rules for epiderivatives of compositions of nonsmooth functions.

Research paper thumbnail of On Relations Between Vector Optimization Problems and Vector Variational Inequalities

Journal of Optimization Theory and Applications

ABSTRACT We obtain equivalences between weak Pareto solutions of vector optimization problems and... more ABSTRACT We obtain equivalences between weak Pareto solutions of vector optimization problems and solutions of vector variational inequalities involving generalized directional derivatives.

Research paper thumbnail of Weak sharp minima: characterizations and su cient conditions

Siam Journal on Control and Optimization, 1999

Research paper thumbnail of An Epigraph-Based Approach to Sensitivity Analysis in Set-Valued Optimization

Springer Proceedings in Mathematics & Statistics, 2013

ABSTRACT In this paper, we obtain estimates for the contingent and adjacent derivatives of the ep... more ABSTRACT In this paper, we obtain estimates for the contingent and adjacent derivatives of the epigraph of the marginal multifunction in parametric set-valued optimization. These estimates generalize some sensitivity results from scalar-valued optimization and provide new information in the setting of multiobjective nonlinear programming.

Research paper thumbnail of Optimization with data perturbations II. The papers are a tribute to Prof. Anthony V. Fiacco on the occasion of his 70th birthday

Annals of Operations Research

The articles of this volume will be reviewed individually. The editors present a short biography ... more The articles of this volume will be reviewed individually. The editors present a short biography and a list of selected publications of A. V. Fiacco. For Vol. I. see ibid. 27 (1990).

Research paper thumbnail of Isotone tangent cones and nonsmooth optimization

Optimization, 1987

A simple but quite general method of formulating necessary optimality conditions for an abstract ... more A simple but quite general method of formulating necessary optimality conditions for an abstract mathematical program is presented. This method centers around tangent cones which are isotone with respect to inclusion. The optimality conditions derived by this ...

Research paper thumbnail of Upper Subderivatives and Generalized Gradients of the Marginal Function of a Non-Lipschitzian Program

Annals of Operations Research - Annals OR, 2001

We obtain an upper bound for the upper subderivative of the marginal function of an abstract para... more We obtain an upper bound for the upper subderivative of the marginal function of an abstract parametric optimization problem when the objective function is lower semicontinuous. Moreover, we apply the result to a nonlinear program with right-hand side perturbations. As a result, we obtain an upper bound for the upper subderivative of the marginal function of a nonlinear program with right-hand side perturbations, which is expressed in “dual form” in terms of appropriate Lagrange multipliers. Finally, we present conditions which imply that the marginal function is locally Lipschitzian.

Research paper thumbnail of Corrigendum to "Convex Subcones of the Contingent Cone in Nonsmooth Calculus and Optimization

Transactions of the American Mathematical Society, 1989

Research paper thumbnail of Dini Derivatives of the Marginal Function of a Non-Lipschitzian Program

SIAM Journal on Optimization, 1996

ABSTRACT Upper and lower bounds are establised for the Dini directional derivatives of the margin... more ABSTRACT Upper and lower bounds are establised for the Dini directional derivatives of the marginal function of a parametric mathematical program. In this program, the equality constraint functions are assumed to be strictly differentiable, but the objective and inequality constraint functions can belong to a large class of non-Lipschitzian functions. A nonsmooth version of the Mangasarian-Fromovitz constraint qualification is also assumed. The main tool in the proofs of these bounds is the calculus of tangent cones.

Research paper thumbnail of Nonsmooth Calculus in Finite Dimensions

SIAM Journal on Control and Optimization, 1987

The notion of subgradient, originally defined for convex functions, has in recent years been exte... more The notion of subgradient, originally defined for convex functions, has in recent years been extended, via the "upper subderivative," to cover functions that are not necessarily convex or even continuous. A number of calculus rules have been proven for these generalized subgradients. This paper develops the finite-dimensional generalized subdifferential calculus for (strictly) lower semicontinuous functions under considerably weaker hypotheses than those previously used. The most general finite-dimensional convex subdifferential calculus results are recovered as corollaries. Other corollaries given include new necessary conditions for optimality in a nonsmooth mathematical program. Various chain rule formulations are considered. Equality in the subdifferential calculus formulae is proven underhypotheses weaker than the usual "subdifferential regularity" assumptions.

Research paper thumbnail of A constraint qualification in quasidifferentiable programming

Optimization, 1991

ABSTRACT One goal in quasid if Terentiable optimization is the development of optimality conditio... more ABSTRACT One goal in quasid if Terentiable optimization is the development of optimality conditions whose hypotheses are independent of the particular choice of quasidifferentials. One such hypothesis, introduced by Demyanov and Rubinov, involves the concept of a pair of convex sets being “in a general position”. In this paper, a simple condition that implies the general position hypothesis is presented. This condition is also shown to be a constraint qualification for non-asymptotic Kuhn-Tucker conditions for a quasidifferentiable program.