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KKT-based primal-dual exactness conditions for the Shor relaxation

JOURNAL OF GLOBAL OPTIMIZATION 2023年第2期86卷 285-301页

作者： Locatelli, M. Univ Parma Dipartimento Ingn & Architettura Parma Italy

In this work we present some exactness conditions for the Shor relaxation of diagonal (or, more generally, diagonalizable) QCQPs, which extend the conditions introduced in different recent papers about the same topic. It is shown that the Shor relaxation is equivalent to two convex quadratic relaxations. Then, sufficient conditions for the exactness of the relaxations are derived from their KKT systems. It will be shown that, in some cases, by this derivation previous conditions in the literature, which can be viewed as dual conditions, since they only involve the Lagrange multipliers appearing in the KKT systems, can be extended to primal-dual conditions, which also involve the primal variables appearing in the KKT systems.

关键词： quadratically constrained quadratic programming Shor relaxation Convex relaxations Exactness conditions

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Convex hull results on quadratic programs with non-intersecting constraints

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MATHEMATICAL programming 2024年第1-2期205卷 539-558页

作者： Joyce, Alexander Yang, Boshi Clemson Univ Sch Math & Stat Sci Clemson SC 29634 USA

Let F subset of R-n be a nonempty closed set. Understanding the structure of the closed convex hull (C) over bar (F) := {( x, xx(T))|x is an element of F} in the lifted space is crucial for solving quadratic programs ... 详细信息

Let F subset of R-n be a nonempty closed set. Understanding the structure of the closed convex hull (C) over bar (F) := <(conv{over bar>{( x, xx(T))|x is an element of F} in the lifted space is crucial for solving quadratic programs related to F. This paper discusses the relationship between C(F) and (C) over bar (G), where G results by adding non-intersecting quadratic constraints to F. We prove that C(G) can be represented as the intersection of C(F) and half spaces defined by the added constraints. The proof relies on a complete description of the asymptotic cones of sets defined by a single quadratic equality and a partial characterization of the recession cone of C(G). Our proof generalizes an existing result for bounded quadratically defined F with non-intersecting hollows and several results on C(G) for G defined by non-intersecting quadratic constraints. The result also implies a sufficient condition for when the lifted closed convex hull of an intersection equals the intersection of the lifted closed convex hulls.

关键词： Convex hull Non-intersecting Semidefinite programming Asymptotic cone quadratically constrained quadratic programming

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Reconfigurable Intelligent Surface-Assisted Multi-User Secrecy Transmission With Low-Resolution DACs

IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING

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IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING 2024年第3期8卷 1205-1221页

作者： Li, Kexin Du, Huiqin Li, Si Jinan Univ Coll Informat Sci & Technol Guangzhou 510632 Peoples R China State Grid Shangrao Power Supply Co State Grid Corp China Beijing 334000 Peoples R China China Southern Power Grid Co Ltd Guangzhou 510630 Peoples R China

This paper considers a reconfigurable intelligent surface (RIS)-assisted multi-user secrecy transmission in the presence of low-resolution digital-to-analog converters (DACs) at a small-cell base station (SBS). The weighted sum secrecy rate (WSSR) is maximized by jointly designing the active beamforming and RIS reflecting phase shift subject to the transmit power and the phase unit-modulus constraints. However, the problem involves two sum-of-logarithms and highly coupled optimization variables. To tackle the non-convex fractional programming problem with multiple ratios, we employ a lower linearization approach for logarithm subtraction and decompose the problem into two quadratically constrained quadratic programming subproblems. The optimum active beamforming is determined using a semi-definite relaxation method, and the a closed-form solution of RIS phase shift matrix is derived through the alternating direction method of multiplier. Moreover, considering practical finite-capacity backhaul link, we develop the user scheduling strategy using the power of transmit beamforming as a discrete indicator and formulate the user scheduling as a mixed-integer constraint. The joint optimization of user scheduling and WSSR is investigated by maximizing the network utility with a l(1)-norm constraint. Simulation results demonstrate the effectiveness of the proposed algorithm in achieving significant WSSR performance even in the presence of low-resolution DACs. Furthermore, these results show that the joint optimization of WSSR and user scheduling can maximize the network utility by selecting the activated subset of served users.

关键词： Array signal processing Optimization Backhaul networks Quantization (signal) Hardware Communication system security programming Reconfigurable intelligent surface (RIS) digital-to-analog converters (DACs) quadratically constrained quadratic programming

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A very simple SQCQP method for a class of smooth convex constrained minimization problems with nice convergence results

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MATHEMATICAL programming 2013年第1-2期142卷 349-369页

作者： Auslender, Alfred Univ Lyon 1 Inst Camille Jordan F-69365 Lyon France Ecole Polytech Dept Econ F-75230 Paris France

We introduce a new and very simple algorithm for a class of smooth convex constrained minimization problems which is an iterative scheme related to sequential quadratically constrained quadratic programming methods, called sequential simple quadratic method (SSQM). The computational simplicity of SSQM, which uses first-order information, makes it suitable for large scale problems. Theoretical results under standard assumptions are given proving that the whole sequence built by the algorithm converges to a solution and becomes feasible after a finite number of iterations. When in addition the objective function is strongly convex then asymptotic linear rate of convergence is established.

关键词： Smooth convex constrained minimization quadratically constrained quadratic programming Sequential quadratic programming Convergence rate Optimal projected gradient schemes

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Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs

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MATHEMATICAL programming 2020年第1期181卷 1-17页

作者： Burer, Samuel Ye, Yinyu Univ Iowa Dept Management Sci Iowa City IA 52242 USA Stanford Univ Inst Computat & Math Engn Dept Management Sci & Engn Stanford CA 94305 USA

We study a class of quadratically constrained quadratic programs (QCQPs), called diagonal QCQPs, which contain no off-diagonal terms x j xk for j = k, and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complements and refines those already present in the literature and can be checked in polynomial time. We then extend our analysis from diagonal QCQPs to general QCQPs, i.e., ones with no particular structure. By reformulating a generalQCQPinto diagonal form, we establish new, polynomial-timecheckable sufficient conditions for the semidefinite relaxations of general QCQPs to be exact. Finally, these ideas are extended to show that a class of random general QCQPs has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables. To the best of our knowledge, this is the first result establishing the exactness of the semidefinite relaxation for random general QCQPs.

关键词： quadratically constrained quadratic programming Semidefinite relaxation Low-rank solutions

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Propagating Uncertainty in Power-System DAE Models With Semidefinite programming

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IEEE TRANSACTIONS ON POWER SYSTEMS 2017年第4期32卷 3146-3156页

作者： Choi, Hyungjin Seiler, Peter J. Dhople, Sairaj V. Univ Minnesota Dept Elect & Comp Engn Minneapolis MN 55455 USA Univ Minnesota Dept Aerosp Engn & Mech Minneapolis MN 55455 USA

This paper outlines a convex-optimization-based method to estimate maximum and minimum bounds on states of differential algebraic equations (DAEs) that describe the electromechanical dynamics of power systems while acknowledging parametric and input uncertainty in the model. The method is based on a second-order Taylor-series approximation of the DAE-model state trajectories as a function of the uncertainties. A key contribution in this regard is the derivation of a DAE model that governs the second-order trajectory sensitivities of states to uncertainties in the model. Bounds on the states are then obtained by solving semidefinite programs, where the objective is to maximize/minimize the Taylor-series approximations subject to constraints that describe the uncertainty space. While the computed bounds are approximate (since they are derived from a Taylor-series approximation of the state trajectories) the method nevertheless is an efficient system-theoretic approach to uncertainty propagation for power-system DAE models. Numerical case studies are presented for a DAE model of the IEEE 39-bus New England system to demonstrate scalability and validate the approach.

关键词： quadratically constrained quadratic programming semidefinite programming trajectory sensitivity analysis uncertainty propagation

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Simultaneous Diagonalization Under Weak Regularity and a Characterization

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第1期203卷 629-650页

作者： Flores-Bazan, Fabian Opazo, Felipe Univ Concepcion Dept Ingn Matemat Casilla 160-C Concepcion Chile

We analyze the fulfillment of the simultaneous diagonalization (SD via congruence) property for any two real matrices, and develop sufficient conditions expressed in different way to those appeared in the last few years. These conditions are established under a different perspective, and in any case, they supplement and clarify other similar results published elsewhere. Following our point of view reflected in a previous work, we offer some necessary and sufficient conditions, different in nature to those in Jiang and Li (SIAM J Optim 26:1649-1668, 2016), for SD: roughly speaking our approach is more geometric and needs to compute images and kernels of matrices;whereas that in Jiang and Li (SIAM J Optim 26:1649-1668, 2016) requires to compute determinant and canonical forms. The bidimensional situation is particularly analyzed, providing new more precise characterizations than those in higher dimension and joint those given earlier by the authors. In addition, we also establish the connection of our characterization of SD with that provided in Jiang and Li (SIAM J Optim 26:1649-1668, 2016).

关键词： Simultaneous diagonalization Congruence quadratically constrained quadratic programming

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Maximizing perturbation radii for robust convex quadratically constrained quadratic programs

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2021年第1期293卷 50-64页

作者： Yu, Pengfei Gao, Ruotian Xing, Wenxun Tsinghua Univ Dept Math Sci Beijing 100084 Peoples R China State Grid Corp China Big Data Ctr Beijing 100052 Peoples R China

Under the assumption that uncertain coefficients corresponding to each constraint are perturbed in an ellipsoidal set, we consider the problem of maximizing the perturbation radius of the ellipsoidal set associated to a robust convex quadratically constrained quadratic programming problem to maintain some properties of a pre-decision. To this end, a fractional programming problem is first formulated to solve the problem, and then equivalently reformulated into linear conic programs over positive semi-definite, second-order cones that are solvable in polynomial time. Numerical experiments in connection with the robust Markowitz's portfolio selection problem are provided to demonstrate the proposed concept of sensitivity analysis. Additionally, certain numerical results are also presented to compare the efficiency of direct solutions of the proposed linear conic programs with that of a bisection method for the corresponding fractional programming problem. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Robustness and sensitivity analysis quadratically constrained quadratic programming Convex programming Linear conic programming

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SOCP reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices

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MATHEMATICAL programming 2018年第2期169卷 531-563页

作者： Jiang, Rujun Li, Duan Wu, Baiyi Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China Guangdong Univ Foreign Studies Sch Finance Guangzhou Guangdong Peoples R China

We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic inequality constraint. By applying a simultaneous block diagonalization approach, we obtain a congruent canonical form for the symmetric matrices in both the objective and constraint functions. By exploiting the block separability of the canonical form, we show that all GTRSs with an optimal value bounded from below are second order cone programming (SOCP) representable. Our result generalizes the recent work of Ben-Tal and den Hertog (Math. Program. 143(1-2):1-29, 2014), which establishes the SOCP representability of the GTRS under the assumption of the simultaneous diagonalizability of the two matrices in the objective and constraint functions. We then derive a closed-form solution for the GTRS when the two matrices are not simultaneously diagonalizable. We further extend our method to two variants of the GTRS in which the inequality constraint is replaced by either an equality constraint or an interval constraint.

关键词： Trust region subproblem Canonical form quadratically constrained quadratic programming

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Constructing Multiple Kernel Learning Framework for Blast Furnace Automation

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IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING 2012年第4期9卷 763-777页

作者： Jian, Ling Gao, Chuanhou Xia, Zhonghang China Univ Petr Coll Sci Qingdao 266555 Peoples R China Dalian Univ Technol Sch Math Sci Dalian 116024 Peoples R China Zhejiang Univ Dept Math Hangzhou 310027 Zhejiang Peoples R China Western Kentucky Univ Dept Comp Sci Bowling Green KY 42101 USA

This paper constructs the framework of the reproducing kernel Hilbert space for multiple kernel learning, which provides clear insights into the reason that multiple kernel support vector machines (SVM) outperform single kernel SVM. These results can serve as a fundamental guide to account for the superiority of multiple kernel to single kernel learning. Subsequently, the constructed multiple kernel learning algorithms are applied to model a nonlinear blast furnace system only based on its input-output signals. The experimental results not only confirm the superiority of multiple kernel learning algorithms, but also indicate that multiple kernel SVM is a kind of highly competitive data-driven modeling method for the blast furnace system and can provide reliable indication for blast furnace operators to take control actions. Note to Practitioners-This paper is motivated by the problem of predicting the silicon content in blast furnace hot metal, which is an open problem for realizing blast furnace automation. Here, based on the single kernel and multiple kernel SVM, we pay special attention to the silicon trend prediction since it can provide more direct guideline for taking control action in the blast furnace operation. Theoretically, we have given the detailed reasons that multiple kernel SVM is superior to single kernel SVM, which can improve the transparency of multiple kernel learning algorithm. The experimental results, not only confirm the superiority of multiple kernel learning algorithms, but also indicate that multiple kernel SVM is a kind of highly competitive data-driven modeling method for the blast furnace system and can provide reliable indication for blast furnace operators to take control actions.

关键词： Data-driven multiple kernel support vector machines (SVM) nonlinear blast furnace system quadratically constrained quadratic programming reproducing kernel Hilbert space

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