Author
Alexey F. Izmailov
Other affiliations: Peoples' Friendship University of Russia, Tambov State University, Russian Academy of Sciences
Bio: Alexey F. Izmailov is an academic researcher from Moscow State University. The author has contributed to research in topics: Sequential quadratic programming & Optimization problem. The author has an hindex of 23, co-authored 122 publications receiving 1575 citations. Previous affiliations of Alexey F. Izmailov include Peoples' Friendship University of Russia & Tambov State University.
Papers published on a yearly basis
Papers
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08 Mar 2014
TL;DR: In this paper, the authors introduce the notion of variational problems with non-isolated solutions and introduce the concept of globalization of convergence, which is a generalization of local methods.
Abstract: 1. Elements of optimization theory and variational analysis.- 2. Equations and unconstrained optimization.- 3. Variational problems: local methods.- 4. Constrained optimization: local methods.- 5. Variational problems: globalization of convergence.- 6. Constrained optimization: globalization of convergence.- 7. Degenerate problems with non-isolated solutions.- A. Miscellaneous material.
184 citations
TL;DR: In this paper, it was shown that the stabilized version of the sSQP algorithm is locally superlinearly convergent under the noncritical multiplier assumption, weaker than SOSC employed originally.
Abstract: The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve superlinear convergence in situations when the Lagrange multipliers associated to a solution are not unique. Within the framework of Fischer (Math Program 94:91–124, 2002), the key to local superlinear convergence of sSQP are the following two properties: upper Lipschitzian behavior of solutions of the Karush-Kuhn-Tucker (KKT) system under canonical perturbations and local solvability of sSQP subproblems with the associated primal-dual step being of the order of the distance from the current iterate to the solution set of the unperturbed KKT system. According to Fernandez and Solodov (Math Program 125:47–73, 2010), both of these properties are ensured by the second-order sufficient optimality condition (SOSC) without any constraint qualification assumptions. In this paper, we state precise relationships between the upper Lipschitzian property of solutions of KKT systems, error bounds for KKT systems, the notion of critical Lagrange multipliers (a subclass of multipliers that violate SOSC in a very special way), the second-order necessary condition for optimality, and solvability of sSQP subproblems. Moreover, for the problem with equality constraints only, we prove superlinear convergence of sSQP under the assumption that the dual starting point is close to a noncritical multiplier. Since noncritical multipliers include all those satisfying SOSC but are not limited to them, we believe this gives the first superlinear convergence result for any Newtonian method for constrained optimization under assumptions that do not include any constraint qualifications and are weaker than SOSC. In the general case when inequality constraints are present, we show that such a relaxation of assumptions is not possible. We also consider applying sSQP to the problem where inequality constraints are reformulated into equalities using slack variables, and discuss the assumptions needed for convergence in this approach. We conclude with consequences for local regularization methods proposed in (Izmailov and Solodov SIAM J Optim 16:210–228, 2004; Wright SIAM J. Optim. 15:673–676, 2005). In particular, we show that these methods are still locally superlinearly convergent under the noncritical multiplier assumption, weaker than SOSC employed originally.
69 citations
TL;DR: This paper derives some new special second-order sufficient optimality conditions for switch-off/switch-on constraints and shows that, quite remarkably, these conditions are actually equivalent to the classical/standard second- order sufficient conditions in optimization.
Abstract: We consider a class of optimization problems with switch-off/switch-on constraints, which is a relatively new problem model. The specificity of this model is that it contains constraints that are being imposed (switched on) at some points of the feasible region, while being disregarded (switched off) at other points. This seems to be a potentially useful modeling paradigm, that has been shown to be helpful, for example, in optimal topology design. The fact that some constraints “vanish” from the problem at certain points, gave rise to the name of mathematical programs with vanishing constraints (MPVC). It turns out that such problems are usually degenerate at a solution, but are structurally different from the related class of mathematical programs with complementarity constraints (MPCC). In this paper, we first discuss some known first- and second-order necessary optimality conditions for MPVC, giving new very short and direct justifications. We then derive some new special second-order sufficient optimality conditions for these problems and show that, quite remarkably, these conditions are actually equivalent to the classical/standard second-order sufficient conditions in optimization. We also provide a sensitivity analysis for MPVC. Finally, a relaxation method is proposed. For this method, we analyze constraints regularity and boundedness of the Lagrange multipliers in the relaxed subproblems, derive a sufficient condition for local uniqueness of solutions of subproblems, and give convergence estimates.
68 citations
TL;DR: This work argues that even weak forms of general constraint qualifications that are suitable for convergence of the augmented Lagrangian methods should not be expected to hold and thus special analysis is needed, and shows convergence to stationary points of the problem under an error bound condition for the feasible set.
Abstract: We consider global convergence properties of the augmented Lagrangian methods on problems with degenerate constraints, with a special emphasis on mathematical programs with complementarity constrai...
64 citations
TL;DR: A modified primal-dual optimality system is derived whose solution is locally unique, nondegenerate, and thus can be found by standard Newton-type techniques.
Abstract: We consider equality-constrained optimization problems, where a given solution may not satisfy any constraint qualification but satisfies the standard second-order sufficient condition for optimality. Based on local identification of the rank of the constraints degeneracy via the singular-value decomposition, we derive a modified primal-dual optimality system whose solution is locally unique, nondegenerate, and thus can be found by standard Newton-type techniques. Using identification of active constraints, we further extend our approach to mixed equality- and inequality-constrained problems, and to mathematical programs with complementarity constraints (MPCC). In particular, for MPCC we obtain a local algorithm with quadratic convergence under the second-order sufficient condition only, without any constraint qualifications, not even the special MPCC constraint qualifications.
51 citations
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26 Jun 2018
TL;DR: This work proposes to formulate physical reasoning and manipulation planning as an optimization problem that integrates first order logic, which it calls Logic-Geometric Programming.
Abstract: We propose to formulate physical reasoning and manipulation planning as an optimization problem that integrates first order logic, which we call Logic-Geometric Programming.
218 citations
TL;DR: An acceleration strategy based on the use of variable metrics and of the Majorize–Minimize principle is proposed and the sequence generated by the resulting Variable Metric Forward–Backward algorithm converges to a critical point of G.
Abstract: We consider the minimization of a function G defined on ${ \mathbb{R} } ^{N}$ , which is the sum of a (not necessarily convex) differentiable function and a (not necessarily differentiable) convex function. Moreover, we assume that G satisfies the Kurdyka---?ojasiewicz property. Such a problem can be solved with the Forward---Backward algorithm. However, the latter algorithm may suffer from slow convergence. We propose an acceleration strategy based on the use of variable metrics and of the Majorize---Minimize principle. We give conditions under which the sequence generated by the resulting Variable Metric Forward---Backward algorithm converges to a critical point of G. Numerical results illustrate the performance of the proposed algorithm in an image reconstruction application.
213 citations
TL;DR: Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.
Abstract: Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by a constraint involving a positive parameter that can be decreased to zero. In the penalized formulation, the complementarity constraint appears as a penalty term in the objective. The existence and uniqueness of solutions for these formulations are investigated, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.
199 citations