We study an extension to Krein spaces of the abstract interpolating spline problem in Hilbert spaces, introduced by M. Atteia. This is a quadratically constrained quadratic programming problem, where the objective fun...
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We study an extension to Krein spaces of the abstract interpolating spline problem in Hilbert spaces, introduced by M. Atteia. This is a quadratically constrained quadratic programming problem, where the objective function is not convex, while the equality constraint is sign indefinite. We characterize the existence of solutions and, if there are any, we describe the set of solutions as the union of a family of affine manifolds parallel to a fixed subspace, which depend on the original data.
We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP). We first develop a general conic program relaxation for QCQP base...
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We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP). We first develop a general conic program relaxation for QCQP based on a matrix decomposition scheme and polyhedral (piecewise linear) underestimation. By employing suitable matrix cones, we then show that the convex conic relaxation can be reduced to a semidefinite programming (SDP) problem. In particular, we investigate polyhedral underestimations for several classes of matrix cones, including the cones of rank-1 and rank-2 matrices, the cone generated by the coefficient matrices, the cone of positive semidefinite matrices and the cones induced by rank-2 semidefinite inequalities. We demonstrate that in general the new SDP relaxations can generate lower bounds at least as tight as the best known SDP relaxations for QCQP. Moreover, we give examples for which tighter lower bounds can be generated by the new SDP relaxations. We also report comparison results of different convex relaxation schemes for nonconvex QCQP with convex quadratic/linear constraints, nonconvex quadratic constraints and 0-1 constraints.
To find a global optimal solution to the quadratically constrained quadratic programming problem, we explore the relationship between its Lagrangian multipliers and related linear conic programming problems. This stud...
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To find a global optimal solution to the quadratically constrained quadratic programming problem, we explore the relationship between its Lagrangian multipliers and related linear conic programming problems. This study leads to a global optimality condition that is more general than the known positive semidefiniteness condition in the literature. Moreover, we propose a computational scheme that provides clues of designing effective algorithms for more solvable quadratically constrained quadratic programming problems.
In this paper we study a Class of nonconvex quadratically constrained quadratic programming problems generalized from relaxations of quadratic assignment problems. We show that each problem is polynomially solved. Str...
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In this paper we study a Class of nonconvex quadratically constrained quadratic programming problems generalized from relaxations of quadratic assignment problems. We show that each problem is polynomially solved. Strong duality holds if a redundant constraint is introduced. As an application, a new lower bound is proposed for the quadratic assignment problem.
In this paper, we study a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. We introduce two convex quadratic relaxations (CQRs) and discuss cases, where the problem i...
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In this paper, we study a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. We introduce two convex quadratic relaxations (CQRs) and discuss cases, where the problem is equivalent to exactly one of the CQRs. Particularly, we show that the global optimal solution can be recovered from an optimal solution of the CQRs. Through this equivalence, we introduce new conditions under which the problem enjoys strong Lagrangian duality, generalizing the recent condition in the literature. Finally, under the new conditions, we present necessary and sufficient conditions for global optimality of the problem.
This paper discusses a class of two-block nonconvex smooth optimization problems with nonlinear constraints. Based on a quadratically constrained quadratic programming (QCQP) approximation, an augmented Lagrangian fun...
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This paper discusses a class of two-block nonconvex smooth optimization problems with nonlinear constraints. Based on a quadratically constrained quadratic programming (QCQP) approximation, an augmented Lagrangian function (ALF), and a Lagrangian splitting technique into small-scale subproblems, we propose a novel sequential quadraticprogramming (SQP) algorithm. First, inspired by the augmented Lagrangian method (ALM), we penalize the quadratic equality constraints associated with the QCQP approximation subproblem in its objective by means of the ALF, and then split the resulting subproblem into two small-scale ones, but both of them are not quadraticprogramming (QP) due to the square of the quadratic equality constraints in the objective. Second, by ignoring the three-order infinitesimal arising from the squared term, the two small-scale subproblems are reduced to two standard QP subproblems, which can yield an improved search direction. Third, taking the ALF of the discussed problem as a merit function, the next iterate point is generated by the Armijo line search. As a result, a new SQP method, called QCQP-based splitting SQP method, is proposed. Under suitable conditions, the global convergence, strong convergence, iteration complexity and convergence rate of the proposed method are analyzed and obtained. Finally, preliminary numerical experiments and applications were carried out, and these show that the proposed method is promising. (C) 2020 Elsevier B.V. All rights reserved.
quadraticallyconstrainedquadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that imm...
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ISBN:
(纸本)9783030457716;9783030457709
quadraticallyconstrainedquadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that immediately implies that 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 convex hull result holds whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. 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.
This paper examines the nonconvex quadratically constrained quadratic programming (QCQP) problems using an iterative method. A QCQP problem can be handled as a linear matrix programming problem with a rank-one constra...
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This paper examines the nonconvex quadratically constrained quadratic programming (QCQP) problems using an iterative method. A QCQP problem can be handled as a linear matrix programming problem with a rank-one constraint on the to-be-determined matrix. One of the existing approaches for solving nonconvex QCQPs relaxes the rank one constraint on the unknown matrix into a semidefinite constraint to obtain the bound on the optimal value without finding the exact solution. By reconsidering the rank one matrix, the iterative rank penalty (IRP) method is proposed to gradually approach the rank one constraint. Each iteration of IRP is formulated as a convex problem with semidefinite constraints. Furthermore, an augmented Lagrangian method, called an extended Uzawa algorithm, is developed to solve the sequential problem at each iteration of IRP for improved scalability and computational efficiency. Simulation examples are presented using the proposed method, and comparative results obtained from the other methods are provided and discussed.
With the large scale of distributed energy resources integrated into distribution systems, the development of effective power management for controllable components is becoming an increasingly important concern. This ...
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With the large scale of distributed energy resources integrated into distribution systems, the development of effective power management for controllable components is becoming an increasingly important concern. This paper focuses on solving the optimal power flow problem in a distribution network with a fully decentralized mode that has massive photovoltaics and electric vehicles. First, an optimal power flow model considering photovoltaics and electric vehicles as controllable agents was established to achieve low network loss, low curtailment of photovoltaic resources, and high satisfaction of electric vehicle owners. Second, a novel linearized power flow model was proposed to produce per-node granularity communication. Thus, each node only needed to exchange the voltage message with its neighboring nodes. Then a decentralized quadratically constrained quadratic programming model based on the alternating direction method of multipliers was built to solve the optimal power flow problem. Next, a closed-form iterative solution method for the decentralized optimization was developed to improve the calculation speed for each iteration. Finally, case studies for a real 35-bus distribution system and a real 110-bus distribution system in China were used to verify the effectiveness of the proposed method.
quadratically constrained quadratic programming (QCQP) can be used to determine the best Q-factor for small antennas with constraints on the antenna efficiency. Constraints on the total directivity and a given front-t...
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ISBN:
(纸本)9788890701887
quadratically constrained quadratic programming (QCQP) can be used to determine the best Q-factor for small antennas with constraints on the antenna efficiency. Constraints on the total directivity and a given front-to-back ratio can also be expressed as QCQP. Such problems are non-convex and hence challenging to solve. Their solution gives the best Q-factor available for any antenna within the considered volume. Thus, solutions to this type of problems provide a tool, which before the design can predict the best possible antenna performance within a given volume of a device. It is hence important to investigate methods to solve this class of QCQP problems. In this paper we compare and investigate two relaxation methods, the Lagrangian dual and semidefinite relaxation, to estimate lower bounds on the Q-factor. The former method is here reduced to solving a generalized eigenvalue-problem. Properties of the different relaxation methods are illustrated and compared. We focus in this paper on the Q-factor and its relation to efficiency, as expressed by the dissipation factor. However, these tools also apply to a larger class of problems including constraints on the directivity and other far-field conditions.
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