This paper examines a complex fractional quadratic optimization problem subject to two quadratic constraints. The original problem is transformed into a parametric quadratic programming problem by the well-known class...
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This paper examines a complex fractional quadratic optimization problem subject to two quadratic constraints. The original problem is transformed into a parametric quadratic programming problem by the well-known classical Dinkelbach method. Then a semidefinite and Lagrangian dual optimization approaches are presented to solve the nonconvex parametric problem at each iteration of the bisection and generalized Newton algorithms. Finally, the numerical results demonstrate the effectiveness of the proposed approaches.
The time delay of arrival- (TDOA-) based source localization using a wireless sensor network has been considered in this paper. The maximum likelihood estimate (MLE) is formulated by taking the correlated TDOA noise i...
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The time delay of arrival- (TDOA-) based source localization using a wireless sensor network has been considered in this paper. The maximum likelihood estimate (MLE) is formulated by taking the correlated TDOA noise into account, which is caused by the difference with the TOA of the reference sensor. The global optimal solution is difficult to obtain due to the nonconvex nature of the ML function. We propose an alternative semidefinite programming method, which transforms the original ML problem into a convex one by relaxing nonconvex equalities into convex matrix inequalities. In addition, the source localization algorithm in the presence of sensor location errors and non-line-of-sight (NLOS) observations is developed. Our simulation results demonstrate the potential advantages of the proposed method. Furthermore, the proposed source localization algorithm by taking the NLOS TOA measurements as the constraints of the convex problem can provide a good estimate.
Solutions for the undrained stability of unsupported excavations are important in practice as they can be used for assessing the safety of temporary excavations associated with various civil works. Even though the pre...
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Solutions for the undrained stability of unsupported excavations are important in practice as they can be used for assessing the safety of temporary excavations associated with various civil works. Even though the previous stability solutions of unsupported excavations namely infinitely long trenches and circular excavations are available in the literature, there is a lack of the stability solution of unsupported rectangular ones. In this paper, the lower bound solutions for the undrained stability of unsupported rectangular excavations in non-homogeneous clays are presented for the first time. A three-dimensional lower bound finite element limit analysis is developed in order to investigate the stability of this problem. The undrained shear strength of non-homogeneous clays are considered as a linearly increasing one with depth. The effects of aspect ratios of rectangular excavations, excavated depth ratios, and normalized strength gradients on the stability number of the problem and its associated failure mechanisms are examined parametrically. A new design equation of the problem is also firstly presented for practical use by practising engineers.
We study time-varying semidefinite programs (TV-SDPs), which are semidefinite programs whose data (and solutions) are functions of time. Our focus is on the setting where the data vary polynomially with time. We show ...
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We study time-varying semidefinite programs (TV-SDPs), which are semidefinite programs whose data (and solutions) are functions of time. Our focus is on the setting where the data vary polynomially with time. We show that under a strict feasibility assumption, restricting the solutions to also be polynomial functions of time does not change the optimal value of the TV-SDP. Moreover, by using a Positivstellensatz (positive locus theorem) on univariate polynomial matrices, we show that the best polynomial solution of a given degree to a TV-SDP can be found by solving a semidefinite program of tractable size. We also provide a sequence of dual problems that can be cast as SDPs and that give upper bounds on the optimal value of a TV-SDP (in maximization form). We prove that under a boundedness assumption, this sequence of upper bounds converges to the optimal value of the TV-SDP. Under the same assumption, we also show that the optimal value of the TV-SDP is attained. We demonstrate the efficacy of our algorithms on a maximum-flow problem with time-varying edge capacities, a wireless coverage problem with time-varying coverage requirements, and on biobjective semidefinite optimization where the goal is to approximate the Pareto curve in one shot.
The inverse linear-quadratic optimal control problem is a system identification problem whose aim is to recover the quadratic cost function and hence the closed-loop system matrices based on observations of optimal tr...
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The inverse linear-quadratic optimal control problem is a system identification problem whose aim is to recover the quadratic cost function and hence the closed-loop system matrices based on observations of optimal trajectories. In this paper, the discrete-time, finite-horizon case is considered, where the agents are also assumed to be homogeneous and indistinguishable. The latter means that the agents all have the same dynamics and objective functions and the observations are in terms of "snap shots" of all agents at different time instants, but what is not known is "which agent moved where" for consecutive observations. This absence of linked optimal trajectories makes the problem challenging. We first show that this problem is globally identifiable. Then, for the case of noiseless observations, we show that the true cost matrix, and hence the closed-loop system matrices, can be recovered as the unique global optimal solution to a convex optimization problem. Next, for the case of noisy observations, we formulate an estimator as the unique global optimal solution to a modified convex optimization problem. Moreover, the statistical consistency of this estimator is shown. Finally, the performance of the proposed method is demonstrated by a number of numerical examples. (c) 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://***/licenses/by-nc-nd/4.0/).
We study a generalization of non-local games-which we call extended non-local games-in which the players, Alice and Bob, initially share a tripartite quantum state with the referee. In such games, the winning conditio...
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We study a generalization of non-local games-which we call extended non-local games-in which the players, Alice and Bob, initially share a tripartite quantum state with the referee. In such games, the winning conditions for Alice and Bob may depend on the outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to randomly selected questions. Our study of this class of games was inspired by the monogamy-of-entanglement games introduced by Tomamichel, Fehr, Kaniewski and Wehner, which they also generalize. We prove that a natural extension of the Navascues-Pironio-Acin hierarchy of semidefinite programmes converges to the optimal commuting measurement value of extended non-local games, and we prove two extensions of results of Tomamichel et al. concerning monogamy-of-entanglement games.
One long-standing challenge in both the optimization and investment communities is to devise an efficient algorithm to select a small number of assets from an asset pool such that a portfolio objective is optimized. T...
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One long-standing challenge in both the optimization and investment communities is to devise an efficient algorithm to select a small number of assets from an asset pool such that a portfolio objective is optimized. This cardinality constrained investment situation naturally arises due to the presence of various forms of market friction, such as transaction costs and management fees, or even due to the consideration of mental cost. Unfortunately, the combinatorial nature of such a portfolio selection problem formulation makes the exact solution process NP-hard in general. We focus in this paper on the cardinality constrained mean-variance portfolio selection problem. Instead of tailoring such a difficult problem into the general solution framework of mixed-integer programming formulation, we explore the special structures and rich geometric properties behind the mathematical formulation. Applying the Lagrangian relaxation to the primal problem results in a pure cardinality constrained portfolio selection problem, which possesses a symmetric property, and to which geometric approaches can be developed. Different from the existing literature that has primarily focused on some direct relaxations of the cardinality constraint, we consider modifying the objective function to some separable relaxations, which are immune to the hard cardinality constraint. More specifically, we develop efficient lower bounding schemes by using the circumscribed box, the circumscribed ball, and the circumscribed axis-aligned ellipsoid to approximate the objective contour of the problem. In particular, all these cardinality constrained relaxation problems can be solved analytically. Furthermore, we derive efficient polynomial-time algorithms for the corresponding dual search problems. Most promisingly, the lower bounding scheme using the circumscribed axis-aligned ellipsoid leads to a semidefinite programming (SDP) formulation and offers a sharp bound and high-quality feasible solution. By i
This paper shows that the alternating direction method of multipliers (ADMM) is efficient for solving the semidefinite inverse quadratic eigenvalue problem (SDIQEP) with a partial eigenstructure. We derive several ADM...
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This paper shows that the alternating direction method of multipliers (ADMM) is efficient for solving the semidefinite inverse quadratic eigenvalue problem (SDIQEP) with a partial eigenstructure. We derive several ADMM-based iterative schemes for SDIQEP and demonstrate their efficiency for large-scale cases of SDIQEP numerically.
Given a linear ordering of the vertices of a graph, the cutwidth of a vertex nu with respect to this ordering is the number of edges from any vertex before nu (including nu) to any vertex after nu in this ordering. Th...
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Given a linear ordering of the vertices of a graph, the cutwidth of a vertex nu with respect to this ordering is the number of edges from any vertex before nu (including nu) to any vertex after nu in this ordering. The cutwidth of an ordering is the maximum cutwidth of any vertex with respect to this ordering. We are interested in finding the cutwidth of a graph, that is, the minimum cutwidth over all orderings, which is an NP-hard problem. In order to approximate the cutwidth of a given graph, we present a semidefinite relaxation. We identify several classes of valid inequalities and equalities that we use to strengthen the semidefinite relaxation. These classes are on the one hand the well-known 3-dicycle equations and the triangle inequalities and on the other hand we obtain inequalities from the squared linear ordering polytope and via lifting the linear ordering polytope. The solution of the semidefinite program serves to obtain a lower bound and also to construct a feasible solution and thereby having an upper bound on the cutwidth. In order to evaluate the quality of our bounds, we perform numerical experiments on graphs of different sizes and densities. It turns out that we produce high quality bounds for graphs of medium size independent of their density in reasonable time. Compared to that, obtaining bounds for dense instances of the same quality is out of reach for solvers using integer linear programming techniques.
As the semidefinite programs that result from integral quadratic constraints are usually large it is important to implement efficient algorithms. The interior-point algorithms in this paper are primal-dual potential r...
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As the semidefinite programs that result from integral quadratic constraints are usually large it is important to implement efficient algorithms. The interior-point algorithms in this paper are primal-dual potential reduction methods and handle multiple constraints. Two approaches are made. For the first approach the computational cost is dominated by a least-squares problem that has to be solved in each iteration. The least-squares problem is solved using an iterative method, namely the conjugate gradient method. The computational effort for the second approach is dominated by forming a linear system of equations. This system of equations is used to compute exact search direction in each iteration. If the number of variables are reduced by solving a small subproblem the resulting system has a very nice structure and can be solved efficiently. The first approach is more efficient for larger problems but is not as numerically stable.
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