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 d...
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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.
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 con...
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We present the expansion of the Basic Sensitivity Theorem to a second order Taylor approach and the implications to explicit model predictive control of quadraticallyconstrained systems. The expansion enables the der...
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We present the expansion of the Basic Sensitivity Theorem to a second order Taylor approach and the implications to explicit model predictive control of quadraticallyconstrained systems. The expansion enables the derivation of an algorithm for the analytical solution of convex multiparametric quadratically constraint programming (mpQCQP) problems and explicit quadraticallyconstrained NMPC problems. We derive the analytical parametric expressions of the control actions for a quadraticallyconstrained MPC problem and its corresponding critical regions. We show the piecewise non-linear form of the solution and closed loop validation of the results. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
In this paper, we present a sequential quadratically constrained quadratic programming (SQCQP) norm-relaxed algorithm of strongly sub-feasible directions for the solution of inequality constrained optimization problem...
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In this paper, we present a sequential quadratically constrained quadratic programming (SQCQP) norm-relaxed algorithm of strongly sub-feasible directions for the solution of inequality constrained optimization problems. By introducing a new unified line search and making use of the idea of strongly sub-feasible direction method, the proposed algorithm can well combine the phase of finding a feasible point (by finite iterations) and the phase of a feasible descent norm-relaxed SQCQP algorithm. Moreover, the former phase can preserve the "sub-feasibility" of the current iteration, and control the increase of the objective function. At each iteration, only a consistent convex quadratically constrained quadratic programming problem needs to be solved to obtain a search direction. Without any other correctional directions, the global, superlinear and a certain quadratic convergence (which is between 1-step and 2-step quadratic convergence) properties are proved under reasonable assumptions. Finally, some preliminary numerical results show that the proposed algorithm is also encouraging. (C) 2009 Published by Elsevier B.V.
In this paper, we consider the extended Celis-Dennis-Tapia (CDT) problem that has a positive duality gap. It is presented in theory that this positive duality gap can be narrowed by adding an appropriate second-order-...
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In this paper, we consider the extended Celis-Dennis-Tapia (CDT) problem that has a positive duality gap. It is presented in theory that this positive duality gap can be narrowed by adding an appropriate second-order-cone (SOC) constraint, which may lead to dividing the problem into two separate subproblems. More concretely, for any extended CDT problem with a positive duality gap, we prove that one SOC constraint is valid to narrow the positive duality gap if and only if the corresponding hyperplane intersects the "open optimal line segment." Especially when the second constraint function consists of the product of two linear functions, we prove that the positive duality gap can be eliminated thoroughly by solving two subproblems with SOC constraints. For any classical CDT problem with a positive duality gap, a new model with two SOC constraints is proposed, and a sufficient condition is presented under which this positive duality gap can be eliminated thoroughly. In particular, based on the sufficient condition, it is proved that the positive duality gaps of any two-dimensional classical CDT problem and a class of three-dimensional classical CDT problems can be eliminated thoroughly. Numerical results of some gap-existing examples coming from other papers show that their positive duality gaps are indeed eliminated by our SOC reformulation technique.
Detecting critical nodes in complex networks (CNP) has great theoretical and practical significance in many disciplines. The existing formulations for CNP are mostly, as we know, based on the integer linear programmin...
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Detecting critical nodes in complex networks (CNP) has great theoretical and practical significance in many disciplines. The existing formulations for CNP are mostly, as we know, based on the integer linear programming model. However, we observed that, these formulations only considered the sizes but neglected the cohesiveness properties of the connected components in the induced network. To solve the problem and improve the performance of CNP solutions, we construct a novel nonconvex quadratically constrained quadratic programming (QCQP) model and derive its approximation solutions via semidefinite programming (SDP) technique and heuristic algorithms. Various types of synthesized and real-world networks, in the context of different connectivity patterns, are used to validate and verify the effectiveness of the proposed model and algorithm. Experimental results show that our method improved the state of the art of the CNP. (C) 2016 Elsevier B.V. All rights reserved.
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 ac...
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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.
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 classi...
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We develop concise primal-dual dynamics for a class of quadratically constrained quadratic programming problems in power system optimization. Using a constrained Lagrangian reformulation of the problem and the classic...
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We develop concise primal-dual dynamics for a class of quadratically constrained quadratic programming problems in power system optimization. Using a constrained Lagrangian 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. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
This paper examines the nonconvex quadratically constrained quadratic programming (QCQP) problems using a decomposition method. It is well known that a QCQP can be transformed into a rank-one constrained optimization ...
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ISBN:
(纸本)9781509059928
This paper examines the nonconvex quadratically constrained quadratic programming (QCQP) problems using a decomposition method. It is well known that a QCQP can be transformed into a rank-one constrained optimization problem. Finding a rank-one matrix is computationally complicated, especially for large scale QCQPs. A decomposition method is applied to decompose the single rank-one constraint on original unknown matrix into multiple rank-one constraints on small scale submatrices. An iterative rank minimization (IRM) algorithm is then proposed to gradually approach all of the rank-one constraints. To satisfy each rank-one constraint in the decomposed formulation, linear matrix inequalities (LMIs) are introduced in IRM with local convergence analysis. The decomposition method reduces the overall computational cost by decreasing size of LMIs, especially when the problem is sparse. Simulation examples with comparative results obtained from an alternative method are presented to demonstrate advantages of the proposed method.
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