Inexact restoration (IR) methods are an important family of numerical methods for solving constra... more Inexact restoration (IR) methods are an important family of numerical methods for solving constrained optimization problems, with applications to electronic structures and bilevel programming, among others areas. In these methods, the minimization is separated into two phases: decreasing infeasibility (feasibility phase) and improving solution (optimality phase). The feasibility phase does not require the generated points to be exactly feasible, so it has a practical appeal. In turn, the optimization phase consists of minimizing a simplified model of the problem over a linearization of the feasible set. In this paper, we introduce a novel optimization phase through a new linearization that carries more information about complementarity than that employed in previous IR strategies. We then prove that the resulting algorithmic scheme is able to converge globally to the so-called complementary approximate KKT points (CAKKT). This global convergence result improves all previous ones for...
... 1, ..., n) i = max j=1:ki i,j = max { i,1, ..., i,ki } , Email address: serkanilter@hotmail.c... more ... 1, ..., n) i = max j=1:ki i,j = max { i,1, ..., i,ki } , Email address: serkanilter@hotmail.com, ilters@istanbul.edu.tr (Serkan Ilter) 1The ... Please be aware that although "Articles in Press" do not have all bibliographic details available yet, they can already be cited using the year of online ...
Sequential optimality conditions for constrained optimization are necessarily satisfied by local ... more Sequential optimality conditions for constrained optimization are necessarily satisfied by local minimizers, independently of the fulfillment of constraint qualifications. These conditions support the employment of different stopping criteria for practical optimization algorithms. On the other hand, when an appropriate property on the constraints holds at a point that satisfies a sequential optimality condition, such a point also satisfies the Karush-Kuhn-Tucker conditions. Those properties will be called strict constraint qualifications in this paper. As a consequence, for each sequential optimality condition, it is natural to ask for its weakest strict associated constraint qualification. This problem has been solved in a recent paper for the Approximate Karush-Kuhn-Tucker sequential optimality condition. In the present paper, we characterize the weakest strict constraint qualifications associated with other sequential optimality conditions that are useful for defining stopping cr...
The constraint nondegeneracy condition is one of the most relevant and useful constraint qualific... more The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, let us say, `-dimensional kernel of the constraint matrix, by the linear independence of a set of `(`+1)/2 derivative vectors. We show that this linear independence requirement can be equivalently formulated in a smaller set, of ` derivative vectors, by considering all orthonormal bases of the kernel instead. This allows us to identify that not all bases are relevant for a constraint qualification to be defined, giving rise to a strictly weaker variant of nondegeneracy related to the global convergence of an external penalty method. Also, by exploiting the sparsity structure of the constraints, we were able to define another weak variant of nondegeneracy by removing the null entries from consideration. In particular, both our new constraint qualifications reduce to the li...
The OVO (Order-Value Optimization) problem consists in the minimization of the order-value functi... more The OVO (Order-Value Optimization) problem consists in the minimization of the order-value function Fp(x), defined by Fp(x) = fip(x)(x), where fi1(x)(x) ≤ . . . ≤ fim(x)(x). The functions f1, . . . , fm are defined on Ω ⊂ IR and p is an integer between 1 and m. When x is a vector of portfolio positions and fi(x) is the predicted loss under the scenario i, the order-value function is the discrete Valueat-Risk (VaR) function, which is largely used in risk evaluations. The OVO problem is continuous but nonsmooth. A Cauchy-like method with guaranteed convergence to points that satisfy a first order optimality condition was recently introduced by Andreani, Dunder ∗Department of Applied Mathematics, IMECC-UNICAMP, University of Campinas, CP 6065, 13081-970 Campinas SP, Brazil. This author was supported by PRONEXOptimization 76.79.1008-00, FAPESP (Grant 01-04597-4) and CNPq (Grant 301115/966). e-mail: andreani@ime.unicamp.br †Department of Applied Mathematics, IMECC-UNICAMP, University of ...
Given F : IR n ! IR m and a closed and convex set, the problem of nding x 2 IR n such that x 2 an... more Given F : IR n ! IR m and a closed and convex set, the problem of nding x 2 IR n such that x 2 and F (x) = 0 is considered. For solving this problem an algorithm of Inexact-Newton type is de-ned. Global and local convergence proofs are presented. As a practical application, the Horizontal Nonlinear Complementarity Problem is introduced. It is shown that the Inexact-Newton algorithm can be applied to this problem. Numerical experiments are performed and commented.
Inexact restoration (IR) methods are an important family of numerical methods for solving constra... more Inexact restoration (IR) methods are an important family of numerical methods for solving constrained optimization problems, with applications to electronic structures and bilevel programming, among others areas. In these methods, the minimization is separated into two phases: decreasing infeasibility (feasibility phase) and improving solution (optimality phase). The feasibility phase does not require the generated points to be exactly feasible, so it has a practical appeal. In turn, the optimization phase consists of minimizing a simplified model of the problem over a linearization of the feasible set. In this paper, we introduce a novel optimization phase through a new linearization that carries more information about complementarity than that employed in previous IR strategies. We then prove that the resulting algorithmic scheme is able to converge globally to the so-called complementary approximate KKT points (CAKKT). This global convergence result improves all previous ones for...
... 1, ..., n) i = max j=1:ki i,j = max { i,1, ..., i,ki } , Email address: serkanilter@hotmail.c... more ... 1, ..., n) i = max j=1:ki i,j = max { i,1, ..., i,ki } , Email address: serkanilter@hotmail.com, ilters@istanbul.edu.tr (Serkan Ilter) 1The ... Please be aware that although "Articles in Press" do not have all bibliographic details available yet, they can already be cited using the year of online ...
Sequential optimality conditions for constrained optimization are necessarily satisfied by local ... more Sequential optimality conditions for constrained optimization are necessarily satisfied by local minimizers, independently of the fulfillment of constraint qualifications. These conditions support the employment of different stopping criteria for practical optimization algorithms. On the other hand, when an appropriate property on the constraints holds at a point that satisfies a sequential optimality condition, such a point also satisfies the Karush-Kuhn-Tucker conditions. Those properties will be called strict constraint qualifications in this paper. As a consequence, for each sequential optimality condition, it is natural to ask for its weakest strict associated constraint qualification. This problem has been solved in a recent paper for the Approximate Karush-Kuhn-Tucker sequential optimality condition. In the present paper, we characterize the weakest strict constraint qualifications associated with other sequential optimality conditions that are useful for defining stopping cr...
The constraint nondegeneracy condition is one of the most relevant and useful constraint qualific... more The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, let us say, `-dimensional kernel of the constraint matrix, by the linear independence of a set of `(`+1)/2 derivative vectors. We show that this linear independence requirement can be equivalently formulated in a smaller set, of ` derivative vectors, by considering all orthonormal bases of the kernel instead. This allows us to identify that not all bases are relevant for a constraint qualification to be defined, giving rise to a strictly weaker variant of nondegeneracy related to the global convergence of an external penalty method. Also, by exploiting the sparsity structure of the constraints, we were able to define another weak variant of nondegeneracy by removing the null entries from consideration. In particular, both our new constraint qualifications reduce to the li...
The OVO (Order-Value Optimization) problem consists in the minimization of the order-value functi... more The OVO (Order-Value Optimization) problem consists in the minimization of the order-value function Fp(x), defined by Fp(x) = fip(x)(x), where fi1(x)(x) ≤ . . . ≤ fim(x)(x). The functions f1, . . . , fm are defined on Ω ⊂ IR and p is an integer between 1 and m. When x is a vector of portfolio positions and fi(x) is the predicted loss under the scenario i, the order-value function is the discrete Valueat-Risk (VaR) function, which is largely used in risk evaluations. The OVO problem is continuous but nonsmooth. A Cauchy-like method with guaranteed convergence to points that satisfy a first order optimality condition was recently introduced by Andreani, Dunder ∗Department of Applied Mathematics, IMECC-UNICAMP, University of Campinas, CP 6065, 13081-970 Campinas SP, Brazil. This author was supported by PRONEXOptimization 76.79.1008-00, FAPESP (Grant 01-04597-4) and CNPq (Grant 301115/966). e-mail: andreani@ime.unicamp.br †Department of Applied Mathematics, IMECC-UNICAMP, University of ...
Given F : IR n ! IR m and a closed and convex set, the problem of nding x 2 IR n such that x 2 an... more Given F : IR n ! IR m and a closed and convex set, the problem of nding x 2 IR n such that x 2 and F (x) = 0 is considered. For solving this problem an algorithm of Inexact-Newton type is de-ned. Global and local convergence proofs are presented. As a practical application, the Horizontal Nonlinear Complementarity Problem is introduced. It is shown that the Inexact-Newton algorithm can be applied to this problem. Numerical experiments are performed and commented.
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