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On extensions of the Frank-Wolfe theorems

COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 1999年第1-3期13卷 87-110页

作者： Luo, ZQ Zhang, SZ McMaster Univ Dept Elect & Comp Engn Commun Res Lab Hamilton ON L8S 4L7 Canada Erasmus Univ Inst Econometr Rotterdam Netherlands

In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty.

关键词： convex quadratic system existence of optimal solutions quadratically constrained quadratic programming

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A Strengthened SDP Relaxation for quadratic Optimization Over the Stiefel Manifold

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第1期202卷 320-339页

作者： Burer, Samuel Park, Kyungchan Univ Iowa Dept Business Analyt Iowa City IA 52242 USA

We study semidefinite programming (SDP) relaxations for the NP-hard problem of globally optimizing a quadratic function over the Stiefel manifold. We introduce a strengthened relaxation based on two recent ideas in the literature: (i) a tailored SDP for objectives with a block-diagonal Hessian;and (ii) the use of the Kronecker matrix product to construct SDP relaxations. Using synthetic instances on five problem classes, we show that, in general, our relaxation significantly strengthens existing relaxations, although at the expense of longer solution times.

关键词： quadratically constrained quadratic programming Semidefinite programming Stiefel manifold

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Time-of-Use Tariff Design Under Uncertainty in Price-Elasticities of Electricity Demand: A Stochastic Optimization Approach

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IEEE TRANSACTIONS ON SMART GRID 2013年第4期4卷 2285-2295页

作者： Ferreira, Rafael de Sa Barroso, Luiz Augusto Lino, Priscila Rochinha Carvalho, Martha Martins Valenzuela, Paula PSR Rio De Janeiro Brazil

Time-of-use (ToU) electricity tariffs are currently employed or considered for implementation in many jurisdictions around the world. In ToU modalities, a set of different tariffs for different hours of the day and/or seasons of the year is defined at the beginning of a given horizon, and then kept constant until its end. While designing ToU tariffs, one of the most significant sources of uncertainty to be considered relates to price-elasticities of demand. We propose an approach for ToU tariff design based in quadratically constrained quadratic programming and stochastic optimization techniques, addressing these uncertainties and dealing with various aspects of tariff design from the point of view of the regulator/regulated utility.

关键词： Demand response electricity tariff design quadratically constrained quadratic programming stochastic optimization time-of-use tariffs

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Scalable Optimization Methods for Incorporating Spatiotemporal Fractionation into Intensity-Modulated Radiotherapy Planning

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INFORMS JOURNAL ON COMPUTING 2022年第2期34卷 1240-1256页

作者： Adibi, Ali Salari, Ehsan Wichita State Univ Dept Ind Syst & Mfg Engn Wichita KS 67260 USA

It has been recently shown that an additional therapeutic gain may be achieved if a radiotherapy plan is altered over the treatment course using a new treatment paradigm referred to in the literature as spatiotemporal fractionation. Because of the nonconvex and large-scale nature of the corresponding treatment plan optimization problem, the extent of the potential therapeutic gain that may be achieved from spatiotemporal fractionation has been investigated using stylized cancer cases to circumvent the arising computational challenges. This research aims at developing scalable optimization methods to obtain high-quality spatiotemporally fractionated plans with optimality bounds for clinical cancer cases. In particular, the treatment-planning problem is formulated as a quadratically constrained quadratic program and is solved to local optimality using a constraint-generation approach, in which each subproblem is solved using sequential linear/quadratic programming methods. To obtain optimality bounds, cutting-plane and column-generation methods are combined to solve the Lagrangian relaxation of the formulation. The performance of the developed methods are tested on deidentified clinical liver and prostate cancer cases. Results show that the proposed method is capable of achieving local-optimal spatiotemporally fractionated plans with an optimality gap of around 10%-12% for cancer cases tested in this study. Summary of Contribution: The design of spatiotemporally fractionated radiotherapy plans for clinical cancer cases gives rise to a class of nonconvex and large-scale quadratically constrained quadratic programming (QCQP) problems, the solution of which requires the development of efficient models and solution methods. To address the computational challenges posed by the large-scale and nonconvex nature of the problem, we employ large-scale optimization techniques to develop scalable solution methods that find local-optimal solutions along with optimality bounds. We

关键词： radiotherapy planning spatiotemporal fractionation quadratically constrained quadratic programming

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SOLVING GENERALIZED CDT PROBLEMS VIA TWO-PARAMETER EIGENVALUES

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SIAM JOURNAL ON OPTIMIZATION 2016年第3期26卷 1669-1694页

作者： Sakaue, Shinsaku Nakatsukasa, Yuji Takeda, Akiko Iwata, Satoru Univ Tokyo Dept Math Informat Tokyo 1138656 Japan

We consider solving a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. This problem is a generalization of the Celis-Denis-Tapia (CDT) problem and thus we refer to it as GCDT (Generalized CDT). The CDT problem has been widely studied, but no polynomial-time algorithm was known until Bienstock's recent work. His algorithm solves the CDT problem in polynomial time with respect to the number of bits in data and log is an element of(-1) by admitting an is an element of error in the constraints. The algorithm, however, appears to be difficult to implement. In this paper, we present another algorithm for GCDT, which is guaranteed to find a global solution for almost all GCDT instances (and slightly perturbed ones in some exceptionally rare cases), in exact arithmetic (including eigenvalue computation). Our algorithm is based on the approach proposed by Iwata, Nakatsukasa, and Takeda (2015) for computing the signed distance between overlapping ellipsoids. Our algorithm computes all the Lagrange multipliers of GCDT by solving a two-parameter linear eigenvalue problem, obtains the corresponding KKT points, and finds a global solution as the KKT point with the smallest objective value. In practice, in finite precision arithmetic, our algorithm requires O(n(6) log log u(-1)) computational time, where n is the number of variables and u is the unit roundoff. Although we derive our algorithm under the unrealistic assumption that exact eigenvalues can be computed, numerical experiments illustrate that our algorithm performs well in finite precision arithmetic.

关键词： quadratically constrained quadratic programming nonconvex optimization Celis-Dennis-Tapia problem two-parameter eigenvalue problem

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Strong duality in minimizing a quadratic form subject to two homogeneous quadratic inequalities over the unit sphere

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JOURNAL OF GLOBAL OPTIMIZATION 2020年第1期76卷 121-135页

作者： Van-Bong Nguyen Thi Ngan Nguyen Sheu, Ruey-Lin Tay Nguyen Univ Dept Math Buon Ma Thuot 632090 Vietnam Natl Cheng Kung Univ Dept Math Tainan 70101 Taiwan

In this paper, we study the strong duality for an optimization problem to minimize a homogeneous quadratic function subject to two homogeneous quadratic constraints over the unit sphere, called Problem (P) in this paper. When a feasible (P) fails to have a Slater point, we show that (P) always adopts the strong duality. When (P) has a Slater point, we propose a set of conditions, called "Property J", on an SDP relaxation of (P) and its conical dual. We show that (P) has the strong duality if and only if there exists at least one optimal solution to the SDP relaxation of (P) which fails Property J. Our techniques are based on various extensions of S-lemma as well as the matrix rank-one decomposition procedure introduced by Ai and Zhang. Many nontrivial examples are constructed to help understand the mechanism.

关键词： quadratically constrained quadratic programming CDT problem S-lemma Slater condition Joint numerical range

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A compact variant of the QCR method for quadratically constrained quadratic 0-1 programs

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OPTIMIZATION LETTERS 2014年第4期8卷 1213-1224页

作者： Galli, Laura Letchford, Adam N. Univ Pisa Dipartimento Informat I-56127 Pisa Italy Univ Lancaster Dept Management Sci Lancaster LA1 4YW England

quadratic Convex Reformulation (QCR) is a technique that was originally proposed for quadratic 0-1 programs, and then extended to various other problems. It is used to convert non-convex instances into convex ones, in such a way that the bound obtained by solving the continuous relaxation of the reformulated instance is as strong as possible. In this paper, we focus on the case of quadratically constrained quadratic 0-1 programs. The variant of QCR previously proposed for this case involves the addition of a quadratic number of auxiliary continuous variables. We show that, in fact, at most one additional variable is needed. Some computational results are also presented.

关键词： Combinatorial optimization Semidefinite programming quadratically constrained quadratic programming

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On the tightness of SDP relaxations of QCQPs

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MATHEMATICAL programming 2022年第1期193卷 33-73页

作者： Wang, Alex L. Kilinc-Karzan, Fatma Carnegie Mellon Univ Pittsburgh PA 15213 USA

quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the projected epigraph of the SDP gives the convex hull of the epigraph in the original QCQP. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.

关键词： quadratically constrained quadratic programming Semidefinite program Relaxation Lagrange function Convex hull

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constrained Primal-Dual Dynamics for QCQP with Applications to Economic Dispatch

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IFAC-PapersOnLine 2017年第2期50卷 215-220页

作者： Kim, Munyoung Adegbege, Ambrose A. Electrical and Computer Enginering Department The College of New Jersey EwingNJ08618 United States

We develop concise primal-dual dynamics for a class of quadratically constrained quadratic programming problems in power system optimization. Using a constrained La-grangian reformulation of the problem and the classical stability result of Lyapunov, we establish the asymptotic convergence of the primal-dual dynamics. We demonstrate the efficiency of the proposed method on an economic power dispatch problem with transmission losses and we suggest a neural network architecture for real-time optimization. © 2017

关键词： Electric load dispatching constrained optimization Dynamics Network architecture quadratic programming Scheduling Asymptotic convergence Economic Dispatch Power system optimization Primal dual quadratically constrained quadratic programming Real time optimization Stability results Transmission loss

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Distributed Quasi-Nonlinear Model Predictive Control with Contractive Constraint

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IFAC-PapersOnLine 2018年第20期51卷 41-47页

作者： Grancharova, Alexandra Johansen, Tor A. Department of Industrial Automation University of Chemical Technology and Metallurgy Bul. Kl. Ohridski 8 Sofia1756 Bulgaria Center for Autonomous Marine Operations and Systems Department of Engineering Cybernetics Norwegian University of Science and Technology Trondheim7491 Norway

An approach to low complexity distributed MPC of nonlinear interconnected systems with coupled dynamics subject to both state and input constraints is proposed. It is based on the idea of introducing a contractive constraint in the centralized NMPC problem formulation, which would guarantee the closed-loop system stability when using a small prediction horizon. Particularly, the one step ahead NMPC problem is considered. Further, a quasi-NMPC method is developed, which is based on a sequential linearization of the nonlinear system dynamics and finding distributedly a suboptimal solution of the resulting convex quadratically constrained quadratic programming problem. The suggested approach would be appropriate for distributed convex NMPC of some cyber-physical systems, since it will reduce the complexity of the on-line NMPC computations, simplify the software implementation, and reduce the requirements for available memory. The proposed method is illustrated with simulations on the model of a quadruple-tank system. © 2018

关键词： Distributed computer systems Closed loop systems Distributed parameter control systems Embedded systems Large scale systems Model predictive control Nonlinear systems Predictive control systems quadratic programming System stability Constraints Distributed control Non linear system dynamics Nonlinear interconnected systems Predictive control quadratically constrained quadratic programming Sequential linearization State and input constraints

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