Regina Burachik - Profile on Academia.edu (original) (raw)

Papers by Regina Burachik

Fenchel duality Epigraph of a convex function Conjugate function a b s t r a c t We establish dua... more Fenchel duality Epigraph of a convex function Conjugate function a b s t r a c t We establish duality results for the generalized monotropic programming problem in separated locally convex spaces. We formulate the generalized monotropic programming (GMP) as the minimization of a (possibly infinite) sum of separable proper convex functions, restricted to a closed and convex cone. We obtain strong duality under a constraint qualification based on the closedness of the sum of the epigraphs of the conjugates of the convex functions. When the objective function is the sum of finitely many proper closed convex functions, we consider two types of constraint qualifications, both of which extend those introduced in the literature. The first constraint qualification ensures strong duality, and is equivalent to the one introduced by Boţ and Wanka. The second constraint qualification is an extension of Bertsekas' constraint qualification and we use it to prove zero duality gap.

The joint EUROPT-OMS conference on optimization, Prague, Czech Republic, July 4–7, 2007, Part I

Optimization Methods and Software

The joint EUROPT-OMS conference on optimization, Prague, Czech Republic, July 4–7, 2007. Part II

Optimization Methods and Software

Journal of Global Optimization, 2006

We study convergence properties of a modified subgradient algorithm, applied to the dual problem ... more We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem.

Generalized proximal point methods for the variational inequality problem

Http Dx Doi Org 10 1080 02331934 2010 527971, Aug 1, 2011

Has Ref. been published yet? If so please provide complete details. Please provide volume and pag... more Has Ref. been published yet? If so please provide complete details. Please provide volume and page numbers for Ref. .

Journal of nonlinear and convex analysis

Set-valued mappings and enlargement of monotone operators

Pacific Journal of Mathematics

Given a maximal monotone operator T in a Banach space, a family of enlargements E(T) of T has bee... more Given a maximal monotone operator T in a Banach space, a family of enlargements E(T) of T has been introduced by Svaiter. He also defined a sum and a positive scalar multiplication of enlargements. The first aim of this work is to further study the properties of these operations. Burachik and Svaiter studied a family of convex functions H(T) which is in a one to one correspondence with E(T). The second

Set-valued mappings & enlargements of monotone operators

This book is addressed to mathematicians, engineers, economists, and researchers interested in ac... more This book is addressed to mathematicians, engineers, economists, and researchers interested in acquiring a solid mathematical foundation in topics such as point-to-set operators, variational inequalities, general equilibrium theory, and nonsmooth optimization, among others. Containing extensive exercises and examples throughout the text, the first four chapters of the book can also be used for a one-quarter course in set-valued analysis and maximal monotone operators for graduate students in pure and applied mathematics, mathematical economics, operations research and related areas. The only requisites, besides a minimum level of mathematical maturity, are some basic results of general topology and functional analysis.

Set-Valued and Variational Analysis, 2015

We introduce a subfamily of additive enlargements of a maximally monotone operator T . Our defini... more We introduce a subfamily of additive enlargements of a maximally monotone operator T . Our definition is inspired by the early work of Fitzpatrick presented in . These enlargements are a subfamily of the family of enlargements E(T ) introduced by Svaiter in . For the case T = ∂f , we prove that some members of the subfamily are smaller than the ε-subdifferential enlargement. In this case, we construct a specific enlargement which coincides with the ε-subdifferential. Since they are all additive, the enlargements in our subfamily can be seen as structurally closer to the ε-subdifferential enlargement. Maximally monotone operator • ε-subdifferential mapping • subdifferential operator • convex lower semicontinuous function • Fitzpatrick function • enlargement of an operator • Brøndsted-Rockafellar enlargements • additive enlargements • Brøndsted-Rockafellar property • Fenchel-Young function.

Enlargements of Monotone Operators

Optimization and Its Applications, 2008

Convex Analysis and Fixed Point Theorems

Optimization and Its Applications, 2008

Set Convergence and Point-to-Set Mappings

Optimization and Its Applications, 2008

Recent Topics in Proximal Theory

Optimization and Its Applications, 2008

ε-Enlargements of Maximal Monotone Operators: Theory and Applications

Applied Optimization, 1998

Springer Proceedings in Mathematics & Statistics, 2013

Enlargements have proven to be useful tools for studying maximally monotone mappings. It is there... more Enlargements have proven to be useful tools for studying maximally monotone mappings. It is therefore natural to ask in which cases the enlargement does not change the original mapping. Svaiter has recently characterized non-enlargeable operators in reflexive Banach spaces and has also given some partial results in the nonreflexive case. In the present paper, we provide another characterization of non-enlargeable operators in nonreflexive Banach spaces under a closedness assumption on the graph. Furthermore, and still for general Banach spaces, we present a new proof of the maximality of the sum of two maximally monotone linear relations. We also present a new proof of the maximality of the sum of a maximally monotone linear relation and a normal cone operator when the domain of the linear relation intersects the interior of the domain of the normal cone.

Springer Optimization and Its Applications, 2010

We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems... more We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems. Our approach is to use a new variant of the deflected subgradient method for solving the dual problem. Our augmented Lagrangian function induces a primal-dual method with strong duality, i.e., with zero duality gap. We prove that our method converges to a dual solution if and only if a dual solution exists. We also prove that all accumulation points of an auxiliary primal sequence are primal solutions. Our results apply, in particular, to classical penalty methods, since the penalty functions associated with these methods can be recovered as a special case of our augmented Lagrangians. Besides the classical augmenting terms given by the 1 -or 2 -norm forms, terms of many other forms can be used in our Lagrangian function. Using a practical selection of the step-size parameters, as well as various choices of the augmenting term, we demonstrate the method on test problems. Our numerical experiments indicate that it is more favourable to use an augmenting term of an exponential form rather than the classical 1 -or 2 -norm forms.

Numerical Algebra, Control and Optimization, 2011

Given a nonconvex and nonsmooth optimization problem, we define a family of "perturbed" Lagrangia... more Given a nonconvex and nonsmooth optimization problem, we define a family of "perturbed" Lagrangians, which induce well-behaved approximations of the dual problem. Our family of approximated problems is said to verify strong asymptotic duality when the optimal dual values of the perturbed problems approach the primal optimal value. Our perturbed Lagrangians can have the same order of smoothness as the functions of the original problem, a property not shared by the classical (unperturbed) augmented Lagrangian. Therefore our proposed scheme allows the use of efficient numerical methods for solving the perturbed dual problems. We establish general conditions under which strong asymptotic duality holds, and we relate the latter with both strong duality and lower semicontinuity of the perturbation function. We illustrate our perturbed duality scheme with two important examples: Constrained Nonsmooth Optimization and Nonlinear Semidefinite programming.

Iterative methods for solving stochastic convex feasibility problems and applications

Fenchel duality Epigraph of a convex function Conjugate function a b s t r a c t We establish dua... more Fenchel duality Epigraph of a convex function Conjugate function a b s t r a c t We establish duality results for the generalized monotropic programming problem in separated locally convex spaces. We formulate the generalized monotropic programming (GMP) as the minimization of a (possibly infinite) sum of separable proper convex functions, restricted to a closed and convex cone. We obtain strong duality under a constraint qualification based on the closedness of the sum of the epigraphs of the conjugates of the convex functions. When the objective function is the sum of finitely many proper closed convex functions, we consider two types of constraint qualifications, both of which extend those introduced in the literature. The first constraint qualification ensures strong duality, and is equivalent to the one introduced by Boţ and Wanka. The second constraint qualification is an extension of Bertsekas' constraint qualification and we use it to prove zero duality gap.

The joint EUROPT-OMS conference on optimization, Prague, Czech Republic, July 4–7, 2007, Part I

Optimization Methods and Software

The joint EUROPT-OMS conference on optimization, Prague, Czech Republic, July 4–7, 2007. Part II

Optimization Methods and Software

Journal of Global Optimization, 2006

We study convergence properties of a modified subgradient algorithm, applied to the dual problem ... more We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem.

Generalized proximal point methods for the variational inequality problem

Http Dx Doi Org 10 1080 02331934 2010 527971, Aug 1, 2011

Has Ref. been published yet? If so please provide complete details. Please provide volume and pag... more Has Ref. been published yet? If so please provide complete details. Please provide volume and page numbers for Ref. .

Journal of nonlinear and convex analysis

Set-valued mappings and enlargement of monotone operators

Pacific Journal of Mathematics

Given a maximal monotone operator T in a Banach space, a family of enlargements E(T) of T has bee... more Given a maximal monotone operator T in a Banach space, a family of enlargements E(T) of T has been introduced by Svaiter. He also defined a sum and a positive scalar multiplication of enlargements. The first aim of this work is to further study the properties of these operations. Burachik and Svaiter studied a family of convex functions H(T) which is in a one to one correspondence with E(T). The second

Set-valued mappings & enlargements of monotone operators

This book is addressed to mathematicians, engineers, economists, and researchers interested in ac... more This book is addressed to mathematicians, engineers, economists, and researchers interested in acquiring a solid mathematical foundation in topics such as point-to-set operators, variational inequalities, general equilibrium theory, and nonsmooth optimization, among others. Containing extensive exercises and examples throughout the text, the first four chapters of the book can also be used for a one-quarter course in set-valued analysis and maximal monotone operators for graduate students in pure and applied mathematics, mathematical economics, operations research and related areas. The only requisites, besides a minimum level of mathematical maturity, are some basic results of general topology and functional analysis.

Set-Valued and Variational Analysis, 2015

We introduce a subfamily of additive enlargements of a maximally monotone operator T . Our defini... more We introduce a subfamily of additive enlargements of a maximally monotone operator T . Our definition is inspired by the early work of Fitzpatrick presented in . These enlargements are a subfamily of the family of enlargements E(T ) introduced by Svaiter in . For the case T = ∂f , we prove that some members of the subfamily are smaller than the ε-subdifferential enlargement. In this case, we construct a specific enlargement which coincides with the ε-subdifferential. Since they are all additive, the enlargements in our subfamily can be seen as structurally closer to the ε-subdifferential enlargement. Maximally monotone operator • ε-subdifferential mapping • subdifferential operator • convex lower semicontinuous function • Fitzpatrick function • enlargement of an operator • Brøndsted-Rockafellar enlargements • additive enlargements • Brøndsted-Rockafellar property • Fenchel-Young function.

Enlargements of Monotone Operators

Optimization and Its Applications, 2008

Convex Analysis and Fixed Point Theorems

Optimization and Its Applications, 2008

Set Convergence and Point-to-Set Mappings

Optimization and Its Applications, 2008

Recent Topics in Proximal Theory

Optimization and Its Applications, 2008

ε-Enlargements of Maximal Monotone Operators: Theory and Applications

Applied Optimization, 1998

Springer Proceedings in Mathematics & Statistics, 2013

Enlargements have proven to be useful tools for studying maximally monotone mappings. It is there... more Enlargements have proven to be useful tools for studying maximally monotone mappings. It is therefore natural to ask in which cases the enlargement does not change the original mapping. Svaiter has recently characterized non-enlargeable operators in reflexive Banach spaces and has also given some partial results in the nonreflexive case. In the present paper, we provide another characterization of non-enlargeable operators in nonreflexive Banach spaces under a closedness assumption on the graph. Furthermore, and still for general Banach spaces, we present a new proof of the maximality of the sum of two maximally monotone linear relations. We also present a new proof of the maximality of the sum of a maximally monotone linear relation and a normal cone operator when the domain of the linear relation intersects the interior of the domain of the normal cone.

Springer Optimization and Its Applications, 2010

We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems... more We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems. Our approach is to use a new variant of the deflected subgradient method for solving the dual problem. Our augmented Lagrangian function induces a primal-dual method with strong duality, i.e., with zero duality gap. We prove that our method converges to a dual solution if and only if a dual solution exists. We also prove that all accumulation points of an auxiliary primal sequence are primal solutions. Our results apply, in particular, to classical penalty methods, since the penalty functions associated with these methods can be recovered as a special case of our augmented Lagrangians. Besides the classical augmenting terms given by the 1 -or 2 -norm forms, terms of many other forms can be used in our Lagrangian function. Using a practical selection of the step-size parameters, as well as various choices of the augmenting term, we demonstrate the method on test problems. Our numerical experiments indicate that it is more favourable to use an augmenting term of an exponential form rather than the classical 1 -or 2 -norm forms.

Numerical Algebra, Control and Optimization, 2011

Given a nonconvex and nonsmooth optimization problem, we define a family of "perturbed" Lagrangia... more Given a nonconvex and nonsmooth optimization problem, we define a family of "perturbed" Lagrangians, which induce well-behaved approximations of the dual problem. Our family of approximated problems is said to verify strong asymptotic duality when the optimal dual values of the perturbed problems approach the primal optimal value. Our perturbed Lagrangians can have the same order of smoothness as the functions of the original problem, a property not shared by the classical (unperturbed) augmented Lagrangian. Therefore our proposed scheme allows the use of efficient numerical methods for solving the perturbed dual problems. We establish general conditions under which strong asymptotic duality holds, and we relate the latter with both strong duality and lower semicontinuity of the perturbation function. We illustrate our perturbed duality scheme with two important examples: Constrained Nonsmooth Optimization and Nonlinear Semidefinite programming.

Iterative methods for solving stochastic convex feasibility problems and applications