Douglas Ward | Miami University (original) (raw)
Papers by Douglas Ward
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 ...
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.
Journal of Mathematical Analysis and Applications, Jul 1, 1995
CRC Press eBooks, Sep 23, 2020
Nonlinear Analysis-theory Methods & Applications, Nov 1, 1988
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.
Systems modelling and optimization, 2022
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.
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.
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.
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.
Siam Journal on Control and Optimization, 1999
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.
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).
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 ...
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.
Transactions of the American Mathematical Society, 1989
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.
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.
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.
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 ...
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.
Journal of Mathematical Analysis and Applications, Jul 1, 1995
CRC Press eBooks, Sep 23, 2020
Nonlinear Analysis-theory Methods & Applications, Nov 1, 1988
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.
Systems modelling and optimization, 2022
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.
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.
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.
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.
Siam Journal on Control and Optimization, 1999
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.
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).
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 ...
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.
Transactions of the American Mathematical Society, 1989
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.
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.
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.