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On local convergence of sequential quadratically-constrained quadratic-programming type methods, with an extension to variational problems

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2008年第2期39卷 143-160页

作者： Fernandez, Damian Solodov, Mikhail Inst Matematica Pura & Aplicada Jardim Bot BR-22460320 Rio De Janeiro Brazil

We consider the class of quadratically-constrained quadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J. Optim. 12, 949-978, 2002) showed superlinear convergence of the primal sequence under the Mangasarian-Fromovitz constraint qualification and the quadratic growth condition. quadratic convergence of the primal-dual sequence was established by Fukushima, Luo and Tseng (SIAM J. Optim. 13, 1098-1119, 2003) under the assumption of convexity, the Slater constraint qualification, and a strong second-order sufficient condition. We obtain a new local convergence result, which complements the above (it is neither stronger nor weaker): we prove primal-dual quadratic convergence under the linear independence constraint qualification, strict complementarity, and a second-order sufficiency condition. Additionally, our results apply to variational problems beyond the optimization case. Finally, we provide a necessary and sufficient condition for superlinear convergence of the primal sequence under a Dennis-More type condition.

关键词： quadratically constrained quadratic programming Karush-Kuhn-Tucker system variational inequality quadratic convergence superlinear convergence Dennis-More condition

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CONVEX HULL PRESENTATION OF A quadratically constrained SET AND ITS APPLICATION IN SOLVING quadratic programming PROBLEMS

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ASIA-PACIFIC JOURNAL OF OPERATIONAL RESEARCH 2009年第6期26卷 769-778页

作者： Xia, Yong Beihang Univ Dept Appl Math LMIB Minist Educ Beijing 100083 Peoples R China Cent Univ Finance & Econ CIAS Beijing 100081 Peoples R China

In this article, we study the convex hull presentation of a quadratically constrained set. Applying the new result, we solve a kind of quadratically constrained quadratic programming problems, which generalizes many w... 详细信息

关键词： Convex hull quadratically constrained quadratic programming

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On the tightness of an SDP relaxation for homogeneous QCQP with three real or four complex homogeneous constraints

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MATHEMATICAL programming 2025年第1-2期211卷 5-48页

作者： Ai, Wenbao Liang, Wei Yuan, Jianhua Beijing Univ Posts & Telecommun Sch Sci Beijing Peoples R China Minist Educ Key Lab Math & Informat Networks Beijing 100876 Peoples R China

In this paper, we consider the problem of minimizing a general homogeneous quadratic function, subject to three real or four complex homogeneous quadratic inequality or equality constraints. For this problem, we present a sufficient and necessary test condition to detect whether its standard semi-definite programming (SDP) relaxation is tight or not. This test condition is based on only an optimal solution pair of the SDP relaxation and its dual. When the tightness is confirmed, a global optimal solution of the original problem is found simultaneously in polynomial-time. While the tightness does not hold, the SDP relaxation and its dual are proved to have the unique optimal solutions. Moreover, the Lagrangian version of such the test condition is specified for non-homogeneous cases. Based on the Lagrangian version, it is proved that several latest sufficient conditions to test the SDP tightness are contained by our test condition under the situation of two constraints. Thirdly, as an application of the test condition, S-lemma and Yuan's lemma are generalized to three real and four complex quadratic forms first under certain exact conditions, which improves some classical results in literature. Finally, a counterexample is presented to show that the test condition cannot be simply extended to four real or five complex homogeneous quadratic constraints.

关键词： quadratically constrained quadratic programming SDP relaxation Rank-one decomposition Tight relaxation Global optimal solutions

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Congressional Apportionment: A Multiobjective Optimization Approach

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MANAGEMENT SCIENCE 2025年第2期71卷 iv-vi, 955页

作者： Shechter, Steven M. Univ British Columbia Sauder Sch Business Vancouver BC V6T 1Z2 Canada

Two events, with major implications for U.S. voters, occur after each decennial census. First, congressional "apportionment" takes place, followed by congressional "districting." Apportionment determines how to allocate the 435 seats in the House of Representatives across the 50 states, whereas districting determines the geographic boundaries assigned to representatives within each state. Although districting and the practice of gerrymandering often receive great attention in the media and courts, the best way to apportion representatives across states has been debated for nearly 250 years. Historical methods (including the current method) each satisfy some desirable optimality criteria that the others are not guaranteed to satisfy. Moreover, none are guaranteed to optimize certain reasonable fairness measures (e.g., minimum range, minimum bias). To our knowledge, we are the first to formulate and analyze a multiobjective optimization approach to apportionment, allowing policymakers to identify Pareto-optimal allocations and quantify their trade-offs between several competing criteria. Some of these models can be formulated and solved as mixed-integer linear programs, whereas others require the solution of mixed-integer, nonconvex, quadratically constrained quadratic programs. We take advantage of recent software advances that allow one to solve these problems with optimality guarantees. Policy implications of our work include Pareto curves from historical censuses and simulations, which suggest opportunities for improvement in some objectives at little sacrifice to others.

关键词： congressional apportionment multiobjective optimization integer programming quadratically constrained quadratic programming

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quadratic programming with one quadratic constraint in Hilbert spaces

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ANNALS OF FUNCTIONAL ANALYSIS 2025年第2期16卷 1-30页

作者： Gonzalez Zerbo, Santiago Maestripieri, Alejandra Martinez Peria, Francisco Consejo Nacl Invest Cient & Tecn Inst Argentino Matemat Alberto P Calderon Saavedra 15Piso 3 RA-1083 Buenos Aires Argentina UNLP Fac Cs Exactas Ctr Matemat Plata Calle 47 Esq 115 RA-1900 La Plata Argentina

A quadratically constrained quadratic programming problem is considered in a Hilbert space setting, where neither the objective nor the constraint are convex functions. Necessary and sufficient conditions are provided... 详细信息

关键词： quadratically constrained quadratic programming Hilbert spaces Operator pencils

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Multiphase Iterative Algorithm for Mixed-Integer Optimal Control

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JOURNAL OF GUIDANCE CONTROL AND DYNAMICS 2025年第4期48卷 757-770页

作者： Pei, Chaoying You, Sixiong Di, Yu Dai, Ran Purdue Univ W Lafayette IN 47907 USA Missouri Univ Sci & Technol Dept Mech & Aerosp Engn Rolla MO 65409 USA Purdue Univ Sch Aeronaut & Astronaut W Lafayette IN 47907 USA Purdue Univ Dept Stat W Lafayette IN 47907 USA

Mixed-integer optimal control problems (MIOCPs) frequently arise in the domain of optimal control problems (OCPs) when decisions including integer variables are involved. However, existing state-of-the-art approaches for solving MIOCPs are often plagued by drawbacks such as high computational costs, low precision, and compromised optimality. In this study, we propose a novel multiphase scheme coupled with an iterative second-order cone programming (SOCP) algorithm to efficiently and effectively address these challenges in MIOCPs. In the first phase, we relax the discrete decision constraints and account for the terminal state constraints and certain path constraints by introducing them as penalty terms in the objective function. After formulating the problem as a quadratically constrained quadratic programming (QCQP) problem, we propose the iterative SOCP algorithm to solve general QCQPs. In the second phase, we reintroduce the discrete decision constraints to generate the final solution. We substantiate the efficacy of our proposed multiphase scheme and iterative SOCP algorithm through successful application to two practical MIOCPs in planetary exploration missions.

关键词： Optimization Algorithm Powered Descent Guidance Mixed Integer Optimal Control quadratically constrained quadratic programming

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STRONG DUALITY FOR THE CDT SUBPROBLEM: A NECESSARY AND SUFFICIENT CONDITION

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SIAM JOURNAL ON OPTIMIZATION 2009年第4期19卷 1735-1756页

作者： Ai, Wenbao Zhang, Shuzhong Beijing Univ Posts & Telecommun Sch Sci Beijing 100876 Peoples R China Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China

In this paper, we consider the problem of minimizing a nonconvex quadratic function, subject to two quadratic inequality constraints. As an application, such a quadratic program plays an important role in the trust region method for nonlinear optimization;such a problem is known as the Celis, Dennis, and Tapia (CDT) subproblem in the literature. The Lagrangian dual of the CDT subproblem is a semidefinite program (SDP), hence convex and solvable. However, a positive duality gap may exist between the CDT subproblem and its Lagrangian dual because the CDT subproblem itself is nonconvex. In this paper, we present a necessary and sufficient condition to characterize when the CDT subproblem and its Lagrangian dual admits no duality gap (i.e., the strong duality holds). This necessary and sufficient condition is easy verifiable and involves only one (any) optimal solution of the SDP relaxation for the CDT subproblem. Moreover, the condition reveals that it is actually rare to render a positive duality gap for the CDT subproblems in general. Moreover, if the strong duality holds, then an optimal solution for the CDT problem can be retrieved from an optimal solution of the SDP relaxation, by means of a matrix rank-one decomposition procedure. The same analysis is extended to the framework where the necessary and sufficient condition is presented in terms of the Lagrangian multipliers at a KKT point. Furthermore, we show that the condition is numerically easy to work with approximatively.

关键词： quadratically constrained quadratic programming strong Lagrangian duality Celis, Dennis, and Tapia subproblem semidefinite program relaxation

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Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

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OPTIMIZATION METHODS & SOFTWARE 2015年第1期30卷 215-249页

作者： Misener, Ruth Smadbeck, James B. Floudas, Christodoulos A. Princeton Univ Dept Chem & Biol Engn Princeton NJ 08544 USA Univ London Imperial Coll Sci Technol & Med Dept Chem Engn London SW7 2AZ England

The global mixed-integer quadratic optimizer, GloMIQO, addresses mixed-integer quadratically constrained quadratic programs (MIQCQP) to epsilon-global optimality. This paper documents the branch-and-cut framework integrated into GloMIQO 2. Cutting planes are derived from reformulation-linearization technique equations, convex multivariable terms, BB convexifications, and low- and high-dimensional edge-concave aggregations. Cuts are based on both individual equations and collections of nonlinear terms in MIQCQP. Novel contributions of this paper include: development of a corollary to Crama's [Concave extensions for nonlinear 0-1 maximization problems, Math. Program. 61 (1993), pp. 53-60] necessary and sufficient condition for the existence of a cut dominating the termwise relaxation of a bilinear expression;algorithmic descriptions for deriving each class of cut;presentation of a branch-and-cut framework integrating the cuts. Computational results are presented along with comparison of the GloMIQO 2 performance to several state-of-the-art solvers.

关键词： quadratically constrained quadratic programming cutting planes global optimization

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Multiparametric/explicit nonlinear model predictive control for quadratically constrained problems

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JOURNAL OF PROCESS CONTROL 2021年 103卷 55-66页

作者： Pappas, Iosif Diangelakis, Nikolaos A. Pistikopoulos, Efstratios N. Texas A&M Univ Artie McFerrin Dept Chem Engn College Stn TX 77843 USA Texas A&M Univ Texas A&M Energy Inst College Stn TX 77843 USA

Explicit model predictive control is an established methodology for the offline determination of the optimal control policy for linear discrete time-invariant systems with linear constraints. Nevertheless, nonlinearities in the form of quadratic constraints naturally appear in process models or are imposed for stability purposes in model predictive control formulations. In this manuscript, we present the theoretical developments and propose an algorithm for the exact solution of explicit nonlinear model predictive control problems with convex quadratic constraints. Our approach is based on a secondorder Taylor approximation of Fiacco's Basic Sensitivity Theorem, which allows for the existence and the analytic derivation of the optimal control actions. The complete exploration of the parameter space is founded on an active set strategy, which employs a pruning criterion to eliminate infeasible active sets. Based on that, the optimal map of solutions is constructed along with the corresponding control actions. The proposed strategy is applied to an explicit nonlinear model predictive control problem with an ellipsoidal terminal set, and comparisons with approximate solutions are drawn to demonstrate the benefits of the presented approach. Furthermore, as a practical application, the optimal operation of a chemostat in the presence of disturbances is exhibited. (C) 2021 Elsevier Ltd. All rights reserved.

关键词： Nonlinear model predictive control Explicit model predictive control quadratically constrained quadratic programming Multiparametric programming

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Design of a multiple kernel learning algorithm for LS-SVM by convex programming

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NEURAL NETWORKS 2011年第5期24卷 476-483页

作者： Jian, Ling Xia, Zhonghang Liang, Xijun Gao, Chuanhou Zhejiang Univ Dept Math Hangzhou 310027 Peoples R China Western Kentucky Univ Dept Comp Sci Bowling Green KY 42101 USA Dalian Univ Technol Sch Math Sci Dalian 116024 Peoples R China China Univ Petr Sch Math & Computat Sci Dongying 257061 Peoples R China

As a kernel based method, the performance of least squares support vector machine (LS-SVM) depends on the selection of the kernel as well as the regularization parameter (Duan, Keerthi, & Poo, 2003). Cross-validation is efficient in selecting a single kernel and the regularization parameter: however, it suffers from heavy computational cost and is not flexible to deal with multiple kernels. In this paper, we address the issue of multiple kernel learning for LS-SVM by formulating it as semidefinite programming (SDP). Furthermore, we show that the regularization parameter can be optimized in a unified framework with the kernel, which leads to an automatic process for model selection. Extensive experimental validations are performed and analyzed. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Least squares support vector machines Multiple kernel learning Convex optimization Semidefinite programming quadratically constrained quadratic programming

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