Semidefinite Programming Problem Research Articles

Bi-modulus constitutive law assumes that material constants have different values in tension and compression. It is known that finding an equilibrium state of an elastic body consisting of a bi-modulus material is recast as a semidefinite programming problem, which can be solved with a primal-dual interior-point method. As an alternative approach, this paper presents a fast first-order optimization method. Specifically, we propose an accelerated proximal gradient method for solving a minimization problem of the total potential energy. This algorithm is easy to implement, and free from numerical solution of linear equations. Numerical experiments demonstrate that the proposed method outperforms the semidefinite programming approach with a standard solver implementing a primal-dual interior-point method.

In this paper, a three-dimensional target localization problem in widely separated multiple-input multiple-output radars is solved using two specific techniques based on time difference of arrival measurements. These techniques are provided in terms of transmitter and receiver antennas, which are named as technique_t and technique_r, respectively. The localization problem is rewritten as a non-convex optimization problem which is based on a least-squares method without any initial estimation. Therefore, a convex semidefinite programming problem is obtained by utilizing the semidefinite relaxation method for the problem which can be performed via the CVX toolbox. Several simulations are provided to evaluate the positioning accuracy in terms of bi-static range error for 3 and 4 transmitter/receiver antennas, different antenna arrangements, and near/far target. In other simulations, the localization accuracy is evaluated in terms of the empirical cumulative density function of positioning error. The results show that the proposed techniques have better accuracy and performance in different scenarios in comparison with other compared methods. The last simulation also demonstrates that the computational time of the mentioned techniques is 0.69 s which is suitable for real-time processing.

The discrete topology and sizing optimization of frame structures with compliance constraints is studied using a novel approach, which is capable of finding the theoretical lower bounds and high-quality discrete solutions in an efficient manner. The proposed approach works by reformulating the discrete problem as a relaxed semidefinite programming (SDP) problem. This reformulation is made possible by a linear relaxation of the original discrete space and the elimination of the nonconvex equilibrium equation using a semidefinite constraint. A continuous global optimum is first derived using existing solvers and then the discrete solution is discovered by the neighborhood search. Numerical examples are presented, including the sizing optimization of 2-Bay 6-Story frame and 3-Bay 10-Story frame, the topology and sizing optimization of 2-Bay 6-Story braced frame. A topology and sizing example with multiple load cases is also provided. The proposed approach and three other metaheuristic algorithms are used to solve these examples. Theoretical lower bounds for these examples can be efficiently discovered by the proposed approach. For the sizing problems, the discrete solutions by the proposed approach are all better than the other algorithms. For the topology and sizing problems, the proposed approach achieves discrete solutions better than genetic algorithm, but worse than the other metaheuristics. The computational superiority of the proposed approach is validated in all the examples.

In this paper, a methodology to solve the optimal reactive power dispatch (ORPD) in electric power systems (EPS), considering discrete controllers, is proposed. Discrete controllers, such as the tap position of on-load tap changing (OLTC) transformers and switchable reactive shunt compensation, are optimized by the proposed method. A semidefinite relaxation (SDR) of the ORPD problem and a branch-and-bound (B&B) algorithm have been fully deployed. A new formulation is presented for the OLTC transformers to obtain a connected structure of the semidefinite programming (SDP) matrices. The customized B&B algorithm deals with the discrete nature of the binary control variables. Moreover, in order to enhance the convexification, valid inequalities called lifted nonlinear cuts (NLC) are implemented in the SDR. Additionally, a chordal decomposition technique is used to improve the computational performance. Finally, the B&B algorithm is used to solve the mixed-integer semidefinite programming problem. Several benchmarks have been used to show the accuracy and scalability of the proposed method, and convergence analysis shows that near-global optimal solutions are generated with small relaxation gaps.

"Positive semi–definite circulant matrices arise in many important applications. The problem arises in various applications where the data collected in a matrix do not maintain the specified structure as is expected in the original system. The task is to retrieve useful information while maintaining the underlying physical feasibility often necessitates search for a good structured approximation of the data matrix. This paper construct structured circulant positive semi–definite matrix that is nearest to a given data matrix. The problem is converted into a semi–definite programming problem as well as a problem comprising a semi–defined program and second-order cone problem. The duality and optimality conditions are obtained and the primal-dual algorithm is outlined. Some of the numerical issues involved will be addressed including unsymmetrical of the problem. Computational results are presented."

We suppose the existence of an oracle which solves any semidefinite programming (SDP) problem satisfying strong feasibility (i.e. Slater's condition) simultaneously at its primal and dual sides. We note that such an oracle might not be able to directly solve general SDPs even after certain regularization schemes are applied. In this work we fill this gap and show how to use such an oracle to ‘completely solve’ an arbitrary SDP. Completely solving entails, for example, distinguishing between weak/strong feasibility/infeasibility and detecting when the optimal value is attained or not. We will employ several tools, including a variant of facial reduction where all auxiliary problems are ensured to satisfy strong feasibility at all sides. Our main technical innovation, however, is an analysis of double facial reduction, which is the process of applying facial reduction twice: first to the original problem and then once more to the dual of the regularized problem obtained during the first run. Although our discussion is focused on semidefinite programming, the majority of the results are proved for general convex cones.

We propose a convex-optimization-based framework for computation of invariant measures of polynomial dynamical systems and Markov processes, in discrete and continuous time. The set of all invariant measures is characterized as the feasible set of an infinite-dimensional linear program (LP). The objective functional of this LP is then used to single out a specific measure (or a class of measures) extremal with respect to the selected functional such as physical measures, ergodic measures, atomic measures (corresponding to, e.g., periodic orbits) or measures absolutely continuous w.r.t. to a given measure. The infinite-dimensional LP is then approximated using a standard hierarchy of finite-dimensional semidefinite programming problems, the solutions of which are truncated moment sequences, which are then used to reconstruct the measure. In particular, we show how to approximate the support of the measure as well as how to construct a sequence of weakly converging absolutely continuous approximations. As a by-product, we present a simple method to certify the nonexistence of an invariant measure, which is an important question in the theory of Markov processes. The presented framework, where a convex functional is minimized or maximized among all invariant measures, can be seen as a generalization of and a computational method to carry out the so-called ergodic optimization, where linear functionals are optimized over the set of invariant measures. Finally, we also describe how the presented framework can be adapted to compute eigenmeasures of the Perron–Frobenius operator.

We propose a method called Polynomial Quadratic Convex Reformulation (PQCR) to solve exactly unconstrained binary polynomial problems (UBP) through quadratic convex reformulation. First, we quadratize the problem by adding new binary variables and reformulating (UBP) into a non-convex quadratic program with linear constraints (MIQP). We then consider the solution of (MIQP) with a specially-tailored quadratic convex reformulation method. In particular, this method relies, in a pre-processing step, on the resolution of a semi-definite programming problem where the link between initial and additional variables is used. We present computational results where we compare PQCR with the solvers Baron and Scip. We evaluate PQCR on instances of the image restoration problem and the low auto-correlation binary sequence problem from MINLPLib. For this last problem, 33 instances were unsolved in MINLPLib. We solve to optimality 10 of them, and for the 23 others we significantly improve the dual bounds. We also improve the best known solutions of many instances.

In this paper we focus on the dynamic state estimation which harnesses a vast amount of sensing data harvested by multiple parties and recognize that in many applications, to improve collaborations between parties, the estimation procedure must be designed with the awareness of protecting participants’ data and model privacy, where the latter refers to the privacy of key parameters of observation models. We develop a state estimation paradigm for the scenario where multiple parties with data and model privacy concerns are involved. Multiple parties monitor a physical dynamic process by deploying their own sensor networks and update the state estimate according to the average state estimate of all the parties calculated by a cloud server and security module. The paradigm taps additively homomorphic encryption which enables the cloud server and security module to jointly fuse parties’ data while preserving the data privacy. Meanwhile, all the parties collaboratively develop a stable (or optimal) fusion rule without divulging sensitive model information. For the proposed filtering paradigm, we analyze the stabilization and the optimality. First, to stabilize the multi-party state estimator while preserving observation model privacy, two stabilization design methods are proposed. For special scenarios, the parties directly design their estimator gains by the matrix norm relaxation. For general scenarios, after transforming the original design problem into a convex semi-definite programming problem, the parties collaboratively derive suitable estimator gains based on the alternating direction method of multipliers (ADMM). Second, an optimal collaborative gain design method with model privacy guarantees is provided, which results in the asymptotic minimum mean square error (MMSE) state estimation. Finally, numerical examples are presented to illustrate our design and theoretical findings.

A new relaxed variant of interior point method for low-rank semidefinite programming problems is proposed in this paper. The method is a step outside of the usual interior point framework. In anticipation to converging to a low-rank primal solution, a special nearly low-rank form of all primal iterates is imposed. To accommodate such a (restrictive) structure, the first order optimality conditions have to be relaxed and are therefore approximated by solving an auxiliary least-squares problem. The relaxed interior point framework opens numerous possibilities how primal and dual approximated Newton directions can be computed. In particular, it admits the application of both the first- and the second-order methods in this context. The convergence of the method is established. A prototype implementation is discussed and encouraging preliminary computational results are reported for solving the SDP-reformulation of matrix-completion problems.

Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This paper develops a provably correct randomized algorithm for solving large, weakly constrained SDP problems by economizing on the storage and arithmetic costs. Numerical evidence shows that the method is effective for a range of applications, including relaxations of MaxCut, abstract phase retrieval, and quadratic assignment. Running on a laptop equivalent, the algorithm can handle SDP instances where the matrix variable has over $10^{14}$ entries.

This study proposes a new method of achieving geographic variety using cooperative relays in conjunction with IDMA to produce dispersed beams. Throughput in wireless networks is something that can perhaps be improved with this combo. The suggested concept requires a two-pronged approach to communication. Initially, users would communicate their information to relaying, that will amplify the data and deliver it to the desired place. IDMA is a well-known NOMA technique that may reduce the effects of MAI at both relay and end nodes. This interference might happen at any of these two nodes. To maintain the final destination’s QoS, the signal was processed carefully at each relay. This research tries to find the optimal beam formation weights by minimizing transmit power while keeping quality (as assessed by signal-to-noise ratio) high (SINR). However, the power minimization problem is not a convex one, thus semi-definite relaxation is used to transform it into a semi-definite programming (SDP) challenge, which may then be solved using the standard SDP problem solver CVX. The mathematical analysis and simulation experiment of the proposed technique show that performance may be enhanced as determined by the bit error rate.

In this paper, we consider improving the secure performance of multiple-input single-output visible light communication channel in the presence of multiple eavesdroppers with multiple photodiodes. Our goal is to design an optimal artificial-noise (AN) aided transmission strategy to maximize the achievable secrecy rate subject to both sum power constraint and peak amplitude constraint. We consider a joint optimization of the transmit covariance and AN covariance for the non-convex secrecy rate maximization (SRM) problem. In order to solve it, the SRM problem is transformed into a series of single-variable semidefinite programming (SDP) problems without losing any optimality, and a one-dimensional search based algorithm is proposed to handle the converted problem, with polynomial complexity. By exploiting Karush-Kuhn-Tucker conditions of the problem, beamforming is found to be optimal for the confidential information transmission. Simulation results show the superior performance of the proposed AN-aided method compared with two other AN-aided methods and no AN-aided method.

Semidefinite Programming (SDP) is a powerful technique to compute tight lower bounds for Optimal Power Flow (OPF) problems. Even using clique decomposition techniques, semidefinite relaxations are still computationally demanding. However, there are many different clique decompositions for the same SDP problem and they are not equivalent in terms of computation time. In this paper, we propose a new strategy to compute efficient clique decompositions with a clique merging heuristic. This heuristic is based on two different estimates of the computational burden of an SDP problem: the size of the problem and an estimation of a per-iteration cost for a state-of-the-art interior-point algorithm. We compare our strategy with other algorithms on MATPOWER instances and we show a significant decrease in solver time.

Regularisation consists in reducing a given optimisation problem to an equivalent form where certain regularity conditions, which guarantee the strong duality, are fulfilled. In this paper, for linear problems of semidefinite programming (SDP), we propose a regularisation procedure which is based on the concept of an immobile index set and its properties. This procedure is described in the form of a finite algorithm which converts any linear semidefinite problem to a form that satisfies the Slater condition. Using the properties of the immobile indices and the described regularization procedure, we obtained new dual SDP problems in implicit and explicit forms. It is proven that for the constructed dual problems and the original problem the strong duality property holds true.

Semidefinite relaxation method for unified near-Field and far-Field localization by AOA

Robust transmit waveform and receive filter design in the presence of eclipsing loss and signal-dependent interference

False data injection (FDI), could cause severe uneconomic system operation and even large blackout, which is further compounded by the increasingly integrated fluctuating renewable generation. As a commonly conducted type of FDI, load redistribution (LR) attack is judiciously manipulated by attackers to alter the load measurement on power buses and affect the normal operation of power systems. In particular, LR attacks have been proved to easily bypass the detection of state estimation. This paper presents a novel distributionally robust optimization (DRO) for operating transmission systems against cyber-attacks while considering the uncertainty of renewable generation. The FDI imposed by an adversary aims to maximally alter system parameters and mislead system operations while the proposed optimization method is used to reduce the risks caused by FDI. Unlike the worst-case-oriented robust optimization, DRO neglects the extremely low-probability case and thus weakens the conservatism, resulting in more economical operation schemes. To obtain computational tractability, a semidefinite programming problem is reformulated and a constraint generation algorithm is utilized to efficiently solve the original problem in a hierarchical master and sub-problem framework. The proposed method can produce more secure and economic operation for the system of rich renewable under LR attacks, reducing load shedding and operation cost to benefit end customers, network operators, and renewable generation.

Electronic structure calculations, in particular the computation of the ground state energy, lead to challenging problems in optimization. These problems are of enormous importance in quantum chemistry for calculations of properties of solids and molecules. Minimization methods for computing the ground state energy can be developed by employing a variational approach, where the second-order reduced density matrix defines the variable. This concept leads to large-scale semidefinite programming problems that provide a lower bound for the ground state energy. Upper bounds of the ground state energy can be calculated for example with the Hartree-Fock method or numerically more exact for a given basis set by full CI. However, Nakata et al. ( J. Chem. Phys.200111482828292) observed that due to numerical errors the semidefinite solver produced erroneous results with a lower bound significantly larger than the full CI energy. For the LiH, CH-, NH-, OH, OH-, and HF molecules violations within one mhartree were observed. We applied the software VSDP which takes all numerical errors due to floating-point arithmetic operations into consideration. For two test libraries VSDP provides tight rigorous error bounds lower than full CI energies reported with an accuracy of 0.1 to 0.01 mhartree. Only little computation work must be spent in order to compute close rigorous error bounds for the ground state energy.

In this letter, we propose an improved pairs of pilots (POP) design method for channel estimation in offset quadrature amplitude modulation based filter bank multicarrier (OQAM/FBMC) systems. Based on an error analysis of the conventional POP channel estimation method, we first develop a heuristic pilot design in which each pilot pair is inserted at two consecutive subcarriers. Besides, we further propose to build a preamble by quartet of subcarriers and conduct a pilot optimization subject to a pilot power constraint, aiming to suppress the noise effect as much as possible. To obtain its optimal solution, we convert the original non-convex pilot optimization problem into a convex semidefinite programming problem, which is shown to be computationally efficient and readily solvable. Numerical simulations validate the effectiveness and superiority of the proposed pilot design method compared to the conventional POP and interference approximation methods.