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A FLEXIBLE ITERATIVE SOLVER FOR nonconvex, EQUALITY-CONSTRAINED QUADRATIC SUBPROBLEMS

SIAM JOURNAL ON SCIENTIFIC COMPUTING 2015年第4期37卷 A1801-A1824页

作者： Hicken, Jason E. Dener, Alp Rensselaer Polytech Inst Mech Aerosp & Nucl Dept Troy NY 12180 USA

We present an iterative primal-dual solver for nonconvex equality-constrained quadratic optimization subproblems. The solver constructs the primal and dual trial steps from the subspace generated by the generalized Arnoldi procedure used in flexible GMRES (FGMRES). This permits the use of a wide range of preconditioners for the primal-dual system. In contrast with FGMRES, the proposed method selects the subspace solution that minimizes a quadratic penalty function over a trust region. Analysis of the method indicates the potential for fast asymptotic convergence near the solution, which is corroborated by numerical experiments. The results also demonstrate the effectiveness and efficiency of the method in the presence of nonconvexity. Overall, the iterative solver is a promising matrix-free linear algebra kernel for inexact-Newton optimization algorithms and is well-suited to partial differential equation constrained optimization.

关键词： PDE-constrained optimization Krylov subspace methods matrix-free optimization nonconvex programming flexible preconditioning inexact Newton

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An outcome space branch and bound-outer approximation algorithm for convex multiplicative programming

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JOURNAL OF GLOBAL OPTIMIZATION 1999年第4期15卷 315-342页

作者： Benson, HP Univ Florida Coll Business Adm Dept Informat & Decis Sci Gainesville FL 32611 USA

This article presents a new global solution algorithm for Convex Multiplicative programming called the Outcome Space Algorithm. To solve a given convex multiplicative program (P-D), the algorithm solves instead an equivalent quasiconcave minimization problem in the outcome space of the original problem. To help accomplish this, the algorithm uses branching, bounding and outer approximation by polytopes, all in the outcome space of problem (P-D). The algorithm economizes the computations that it requires by working in the outcome space, by avoiding the need to compute new vertices in the outer approximation process, and, except for one convex program per iteration, by requiring for its execution only linear programming techniques and simple algebra.

关键词： multiplicative programming convex multiplicative programming global optimization outer approximation branch and bound nonconvex programming

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On convex transformability and the solution of nonconvex problems via the dual of Falk

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2012年第2期46卷 171-185页

作者： Wood, Derren W. Groenwold, Albert A. Univ Stellenbosch Dept Mech & Mech Engn ZA-7602 Matieland South Africa

In structural optimization, most successful sequential approximate optimization (SAO) algorithms solve a sequence of strictly convex subproblems using the dual of Falk. Previously, we have shown that, under certain conditions, a nonconvex nonlinear (sub)problem may also be solved using the Falk dual. In particular, we have demonstrated this for two nonconvex examples of approximate subproblems that arise in popular and important structural optimization problems. The first is used in the SAO solution of the weight minimization problem, while the topology optimization problem that results from volumetric penalization gives rise to the other. In both cases, the nonconvex subproblems arise naturally in the consideration of the physical problems, so it seems counter productive to discard them in favor of using standard, but less well-suited, strictly convex approximations. Though we have not required that strictly convex transformations exist for these problems in order that they may be solved via a dual approach, we have noted that both of these examples can indeed be transformed into strictly convex forms. In this paper we present both the nonconvex weight minimization problem and the nonconvex topology optimization problem with volumetric penalization as instructive numerical examples to help motivate the use of nonconvex approximations as subproblems in SAO. We then explore the link between convex transformability and the salient criteria which make nonconvex problems amenable to solution via the Falk dual, and we assess the effect of the transformation on the dual problem. However, we consider only a restricted class of problems, namely separable problems that are at least C (1) continuous, and a restricted class of transformations: those in which the functions that represent the mapping are each continuous, monotonic and univariate.

关键词： Falk dual Weight minimization Topology optimization nonconvex programming Convex transformation

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A branch and bound algorithm for nonconvex quadratic optimization with ball and linear constraints

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JOURNAL OF GLOBAL OPTIMIZATION 2017年第2期69卷 309-342页

作者： Beck, Amir Pan, Dror Technion Israel Inst Technol Fac Ind Engn & Management Haifa Israel

We suggest a branch and bound algorithm for solving continuous optimization problems where a (generally nonconvex) objective function is to be minimized under nonconvex inequality constraints which satisfy some specific solvability assumptions. The assumptions hold for some special cases of nonconvex quadratic optimization problems. We show how the algorithm can be applied to the problem of minimizing a nonconvex quadratic function under ball, out-of-ball and linear constraints. The main tool we utilize is the ability to solve in polynomial computation time the minimization of a general quadratic under one Euclidean sphere constraint, namely the so-called trust region subproblem, including the computation of all local minimizers of that problem. Application of the algorithm on sparse source localization problems is presented.

关键词： Quadratically constrained quadratic problems nonconvex programming Branch and bound Sparse source localization Trust region subproblem

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On concave constraint functions and duality in predominantly black-and-white topology optimization

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2010年第33-36期199卷 2224-2234页

作者： Wood, Derren W. Groenwold, Albert A. Univ Stellenbosch Dept Mech Engn ZA-7600 Stellenbosch South Africa

We study the 'classical' discrete, solid-void or black-and-white topology optimization problem, in which minimum compliance is sought, subject to constraints on the available material resource. We assume that this problem is solved using methods that relax the discreteness requirements during intermediate steps, and that the associated programming problems are solved using sequential approximate optimization (SAO) algorithms based on duality. More specifically, we assume that the advantages of the well-known Falk dual are exploited. Such algorithms represent the state-of-the-art in (large-scale) topology optimization when multiple constraints are present: an important example being the method of moving asymptotes (MMA). We depart by noting that the aforementioned SAO algorithms are invariably formulated using strictly convex subproblems. We then numerically illustrate that strictly concave constraint functions, like those present in volumetric penalization, as recently proposed by Bruns and co-workers, may increase the difficulty of the topology optimization problem when strictly convex approximations are used in the SAO algorithm. In turn, volumetric penalization methods are of notable importance, since they seem to hold much promise for generating predominantly solid-void or discrete designs. We then argue that the nonconvex problems we study may in some instances efficiently be solved using dual SAO methods based on nonconvex (strictly concave) approximations which exhibit monotonicity with respect to the design variables. Indeed, for the topology problem resulting from SIMP-like volumetric penalization, we show explicitly that convex approximations are not necessary. Even though the volumetric penalization constraint is strictly concave, the maximum of the resulting dual subproblem still corresponds to the optimum of the original primal approximate subproblem. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Topology optimization SIMP Volumetric penalization Sequential approximate optimization (SAO) nonconvex programming Falk dual

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Subspace-Modeling-Based Nonlinear Measurement for Process Design

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INDUSTRIAL & ENGINEERING CHEMISTRY RESEARCH 2011年第23期50卷 13457-13465页

作者： Lu, XinJiang Huang, MingHui Li, YiBo Chen, Min Cent S Univ State Key Lab High Performance Complex Mfg Changsha 410083 Hunan Peoples R China

In industry, many nonlinear processes can be approximated well by a linear model under a suitable design parameter, upon which a linear controller can effectively control these processes. The key problem is how to find this suitable design parameter, which is never considered in process design. In this paper, a novel design for control approach is proposed to design the process to have a satisfactory linear approximation. First, a subspace-modeling-based nonlinear measurement is proposed to avoid the infinite dimensional optimization problem in the traditional nonlinearity measurement. Then a particle-swarm-optimization-based design approach is developed to obtain the optimal design parameter through solving the nonconvex and nondifferential measurement problem. Finally, the proposed design is applied to design a practical curing oven and compared with the existing design.

关键词： PARTICLE swarm optimization MATHEMATICAL models MATHEMATICAL optimization EXPERIMENTAL design PROBLEM solving nonconvex programming NONLINEAR programming

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Semi-quasidifferentiability in nonsmooth nonconvex multiobjective optimization

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2022年第1期299卷 35-45页

作者： Kabgani, Alireza Soleimani-damaneh, Majid Urmia Univ Technol Appl Math Grp Orumiyeh Iran Inst Res Fundamental Sci IPM Sch Math POB 19395-5746 Tehran Iran Univ Tehran Sch Math Stat & Comp Sci Coll Sci Tehran Iran

In this paper, we extend the concept of quasidifferential to a new notion called semi-quasidifferential. This generalization is motivated by the convexificator notion. Some important properties of semiquasidifferentials are established. The relationship between semi-quasidifferentials and the Clarke sub differential is studied, and a mean value theorem in terms of semi-quasidifferentials is proved. It is shown that this notion is helpful to investigate nonsmooth optimization problems even when the objective and/or constraint functions are discontinuous. Considering a multiobjective optimization problem, a characterization of some cones related to the feasible set is provided. They are used for deriving necessary and sufficient optimality conditions. We close the paper by obtaining optimality conditions in multi objective optimization in terms of semi-quasidifferentials. Some outcomes of the current work generalize the related results existing in the literature. (c) 2021 Elsevier B.V. All rights reserved.

关键词： Multiple objective programming Nonsmooth optimization Semi-quasidifferentiable function nonconvex programming

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Optimality condition and complexity analysis for linearly-constrained optimization without differentiability on the boundary

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MATHEMATICAL programming 2019年第1-2期178卷 263-299页

作者： Haeser, Gabriel Liu, Hongcheng Ye, Yinyu Univ Sao Paulo Inst Math & Stat Dept Appl Math Sao Paulo SP Brazil Stanford Univ Dept Radiat Oncol Stanford CA 94305 USA Stanford Univ Dept Management Sci & Engn Stanford CA 94305 USA

In this paper we consider the minimization of a continuous function that is potentially not differentiable or not twice differentiable on the boundary of the feasible region. By exploiting an interior point technique, we present first- and second-order optimality conditions for this problem that reduces to classical ones when the derivative on the boundary is available. For this type of problems, existing necessary conditions often rely on the notion of subdifferential or become non-trivially weaker than the KKT condition in the (twice-)differentiable counterpart problems. In contrast, this paper presents a new set of first- and second-order necessary conditions that are derived without the use of subdifferential and reduce to exactly the KKT condition when (twice-)differentiability holds. As a result, these conditions are stronger than some existing ones considered for the discussed minimization problem when only non-negativity constraints are present. To solve for these optimality conditions in the special but important case of linearly constrained problems, we present two novel interior point trust-region algorithms and show that their worst-case computational efficiency in achieving the potentially stronger optimality conditions match the best known complexity bounds. Since this work considers a more general problem than those in the literature, our results also indicate that best known existing complexity bounds are actually held for a wider class of nonlinear programming problems. This new development is significant since optimality conditions play a fundamental role in computational optimization and more and more nonlinear and nonconvex problems need to be solved in practice.

关键词： Constrained optimization nonconvex programming Interior point method First order algorithm Nonsmooth problems

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Maximizing a concave function over the efficient or weakly-efficient set

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1999年第2期117卷 239-252页

作者： Horst, R Thoai, NV Univ Trier Dept Math D-54286 Trier Germany

An important approach in multiple criteria linear programming is the optimization of some function over the efficient or weakly-efficient set. This is a very difficult nonconvex optimization problem, even for the case that the function to be optimized is linear. In this article we consider the problem of maximizing a concave function over the efficient or weakly-efficient set. We show that this problem can essentially be formulated as a special global optimization problem in the space of the extreme criteria of the underlying multiple criteria linear program. An algorithm of branch and bound type is proposed for solving the resulting problem. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： multiple criteria analysis optimization over efficient/weakly-effcient sets nonconvex programming global optimization branch and bound

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ON THE CONVERGENCE OF MIRROR DESCENT BEYOND STOCHASTIC CONVEX programming

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SIAM JOURNAL ON OPTIMIZATION 2020年第1期30卷 687-716页

作者： Zhou, Zhengyuan Mertikopoulos, Panayotis Bambos, Nicholas Boyd, Stephen P. Glynn, Peter W. IBM Res Yorktown Hts NY 10598 USA NYU Stern Sch Business 550 1St Ave New York NY 10012 USA Univ Grenoble Alpes CNRS Grenoble INP InriaLIG F-38000 Grenoble France Stanford Univ Dept Elect Engn Dept Management Sci & Engn Stanford CA 94305 USA

In this paper, we examine the convergence of mirror descent in a class of stochastic optimization problems that are not necessarily convex (or even quasi-convex) and which we call variationally coherent. Since the standard technique of "ergodic averaging" offers no tangible benefits beyond convex programming, we focus directly on the algorithm's last generated sample (its "last iterate"), and we show that it converges with probabiility 1 if the underlying problem is coherent. We further consider a localized version of variational coherence which ensures local convergence of Stochastic mirror descent (SMD) with high probability. These results contribute to the landscape of nonconvex stochastic optimization by showing that (quasi-)convexity is not essential for convergence to a global minimum: rather, variational coherence, a much weaker requirement, suffices. Finally, building on the above, we reveal an interesting insight regarding the convergence speed of SMD: in problems with sharp minima (such as generic linear programs or concave minimization problems), SMD reaches a minimum point in a finite number of steps (a.s.), even in the presence of persistent gradient noise. This result is to be contrasted with existing black-box convergence rate estimates that are only asymptotic.

关键词： mirror descent nonconvex programming stochastic optimization stochastic approximation variational coherence

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