China 
Africa 
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Introduction
Published in Yunmin Zhu, Jie Zhou, Xiaojing Shen, Enbin Song, Yingting Luo, Networked Multisensor Decision and Estimation Fusion, 2012
Yunmin Zhu, Jie Zhou, Xiaojing Shen, Enbin Song, Yingting Luo
which is an SDP with variables x∈Rn and a symmetric X variable. It can be shown that the SDP relaxation problem (1.78) is, in essence, the bi-dual of (Equation 1.76) (see Boyd and Vandenberghe (2004)). In contrast to the original QCQP (Equation 1.76), the SDP relaxation problem (1.78) is a convex optimization problem, and thus can be efficiently solved by interior-point methods in polynomial time.
A hybrid approach for transmission grid resilience assessment using reliability metrics and power system local network topology
Published in Sustainable and Resilient Infrastructure, 2021
Binghui Li, Dorcas Ofori-Boateng, Yulia R. Gel, Jie Zhang
In addition, the OPF problem in Equation 4 is formulated with DC power flow constraints (DC-OPF) as opposed to the complete AC form, which is also known as the AC-OPF problem (Wood & Wollenberg, 2012). The simplified DC-OPF model only includes active power flow terms and drops the reactive power flow terms, which results in a set of completely linear power flow constraints and fits in the field of linear or quadratic programming. The DC-OPF model is adopted since global optimality is guaranteed due to convexity. By contrast, the full AC-OPF problem is modeled as a quadratically constrained quadratic programming (QCQP) problem and the feasibility region is typically a non-convex set. While convex QCQP problems can be solved efficiently, it is extremely challenging to solve non-convex QCQP problems (Calafiore & El Ghaoui, 2014). While the accuracy of DC-OPF approximation is questionable in distribution networks, it does not cause systematic errors in large-scale transmission networks, such as the one in this analysis.
A branch-and-cut algorithm using polar cuts for solving nonconvex quadratic programming problems
Published in Optimization, 2018
Zhibin Deng, Shu-Cherng Fang, Cheng Lu, Xiaoling Guo
Most global solution methods for QCQP are based on the convex relaxation of the original problem embedded in a branch-and-bound (or branch-and-cut) framework where a lower or upper bound is computed by some relaxation schemes, such as linearization and second-order cone programming (SOCP). For example, Al-Khayyal et al. [7] proposed a branch-and-bound algorithm based on the linear programming relaxation over an outer polyhedral approximation of QCQP. Linderoth [8] and Raber [9] extended this work and developed a branch-and-bound algorithm involving linear programming subproblems based on a simplicial partition. Sherali and Tuncbilek [10] developed linear programming relaxations based on the reformulation-and-linearization techniques (RLT). Audet et al. [11] extended the use of RLT to solve QCQP by including different classes of linearizations. Kim and Kojima [12] applied the lift-and-project idea of RLT to create a SOCP relaxation for QCQP. Notice that the relaxation problems in all of these works are either linear programming (LP) or SOCP based.
Minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint
Published in Optimization, 2018
S. Fallahi, M. Salahi, T. Terlaky
(QCQP) appears in various disciplines such as: optimal control and system theory [1,2], physics [3,4], chemistry [5–7], engineering [8–10], biology and medicine [11,12], economics and sociology [13]. The known and widely used trust region subproblem in continuous optimization also is a special case of (QCQP) [14,15]. Global optimality conditions for some cases of (QCQP) were developed in [16–19]. In [20] the authors have considered (QCQP) and discussed the attainability of the optimal primal solution without the Slater condition. They have shown that if both the (QCQP) problem and its Lagrangian dual problem are feasible, then there does not exist any duality gap between their optimal values. In [21] the authors have considered a quadratic optimization problem with a single quadratic inequality constraint. They have discussed and compared various assumptions necessary for solving (QCQP) such as Slater condition and simultaneously diagonalizable via congruence. In [18] the authors have examined the relationship between global optimality of inequality constrained non-convex optimization problems and Lagrange multiplier conditions. They have used the S-lemma to derive optimality conditions. In a most recent work, Ben-Tal and Hertog studied quadratic optimization with one and two quadratic constraints [22,23]. They have shown when the quadratic forms are simultaneously diagonalizable, it is possible to derive an equivalent conic quadratic problem, which is significantly more tractable than a semidefinite optimization (SDO) relaxation approach for such classes of problems [24–26]. In [19] Stern and Wolkowicz have studied the two-sided nonconvex quadratically constrained problem