Semidefinite Programming Relaxation Research Articles

Exactness Conditions for Semidefinite Programming Relaxations of Generalization of the Extended Trust Region Subproblem

The extended trust region subproblem (ETRS) of minimizing a quadratic objective over the unit ball with additional linear constraints has attracted a lot of attention in the last few years because of its theoretical significance and wide spectra of applications. Several sufficient conditions to guarantee the exactness of its semidefinite programming (SDP) relaxation have been recently developed in the literature. In this paper, we consider a generalization of the extended trust region subproblem (GETRS), in which the unit ball constraint in the ETRS is replaced by a general, possibly nonconvex, quadratic constraint, and the linear constraints are replaced by a conic linear constraint. We derive sufficient conditions for guaranteeing the exactness of the SDP relaxation of the GETRS under mild assumptions. Our main tools are two classes of convex relaxations for the GETRS based on either a simultaneous diagonalization transformation of the quadratic forms or linear combinations of the quadratic forms. We also compare our results to the existing sufficient conditions in the literature. Finally, we extend our results to derive a new sufficient condition for the exactness of the SDP relaxation of general diagonal quadratically constrained quadratic programs, where each quadratic function is associated with a diagonal matrix. Funding: This work was supported by the National Natural Science Foundation of China [Grants 11801087, 12171100, 72161160340]; Hong Kong Research Grants Council [Grants 14213716 and 14202017]; and Natural Science Foundation of Shanghai [Grant 22ZR1405100].

Linear Programming and Community Detection

The problem of community detection with two equal-sized communities is closely related to the minimum graph bisection problem over certain random graph models. In the stochastic block model distribution over networks with community structure, a well-known semidefinite programming (SDP) relaxation of the minimum bisection problem recovers the underlying communities whenever possible. Motivated by their superior scalability, we study the theoretical performance of linear programming (LP) relaxations of the minimum bisection problem for the same random models. We show that, unlike the SDP relaxation that undergoes a phase transition in the logarithmic average degree regime, the LP relaxation fails in recovering the planted bisection with high probability in this regime. We show that the LP relaxation instead exhibits a transition from recovery to nonrecovery in the linear average degree regime. Finally, we present nonrecovery conditions for graphs with average degree strictly between linear and logarithmic. Funding: A. Del Pia is partially funded by Office of Naval Research (ONR) [Grant N00014-19-1-2322]. D. Kunisky is supported by the ONR [Grant N00014-20-1-2335], a Simons Investigator Award to Daniel Spielman, and the National Science Foundation [Grants DMS-1712730 and DMS-1719545].

An exact algorithm for semi-supervised minimum sum-of-squares clustering

The minimum sum-of-squares clustering (MSSC), or k-means type clustering, is traditionally considered an unsupervised learning task. In recent years, the use of background knowledge to improve the cluster quality and promote interpretability of the clustering process has become a hot research topic at the intersection of mathematical optimization and machine learning research. The problem of taking advantage of background information in data clustering is called semi-supervised or constrained clustering. In this paper, we present branch-and-cut algorithm for semi-supervised MSSC, where background knowledge is incorporated as pairwise must-link and cannot-link constraints. For the lower bound procedure, we solve the semidefinite programming relaxation of the MSSC discrete optimization model, and we use a cutting-plane procedure for strengthening the bound. For the upper bound, instead, by using integer programming tools, we use an adaptation of the k-means algorithm to the constrained case. For the first time, the proposed global optimization algorithm efficiently manages to solve real-world instances up to 800 data points with different combinations of must-link and cannot-link constraints and with a generic number of features. This problem size is about four times larger than the one of the instances solved by state-of-the-art exact algorithms.

Convex relaxation for optimal fixture layout design

This article proposes a general fixture layout design framework that directly integrates the system equation with the convex relaxation method. Note that the optimal fixture design problem is a large-scale combinatorial optimization problem; we relax it to a convex Semi-Definite Programming (SDP) problem by adopting sparse learning and SDP relaxation techniques. It can be solved efficiently by existing convex optimization algorithms and thus generates a near-optimal fixture layout. A real case study in the half-to-half fuselage assembly process indicates the superiority of our proposed algorithm compared to the current industry practice and state-of-art methods.

Optimal power flow schedules with reduced low-frequency oscillations

The dynamic response of power grids to small events or persistent stochastic disturbances influences their stable operation. Low-frequency inter-area oscillations are of particular concern due to insufficient damping. This paper studies the effect of the operating point on the linear time-invariant dynamics of power networks. A pertinent metric based on the frequency response of grid dynamics is proposed to quantify power system’s stability against inter-area oscillations. We further put forth an optimal power flow formulation to yield a grid dispatch that optimizes this novel stability metric. A semidefinite program (SDP) relaxation is employed to yield a computationally tractable convex problem. Numerical tests on the IEEE-39 bus system demonstrate that the SDP relaxation is exact yielding a rank-1 solution. The relative trade-off of the proposed small-signal stability metric versus the generation cost is also studied.

Certified and accurate SDP bounds for the ACOPF problem

We propose a new method for improving the bound tightness of the popular semidefinite programming (SDP) relaxation for the ACOPF introduced in Lavaei and Low (2012), Molzahn and Hiskens (2019). First, we reformulate the ACOPF Lagrangian dual as an unconstrained concave maximization problem with a clique decomposition induced sparse structure. We prove that this new formulation has the same optimal value as the SDP relaxation. We then use the solution of the SDP relaxation as a starting point for a tailored structure-aware bundle method. This post-processing technique significantly improves the tightness of the SDP bounds computed by the state-of-the-art solver MOSEK, as shown by our computational experiments on large-scale instances from PGLib-OPF v21.07. For ten of the tested instances, our post-processing decreases by more than 50% the optimality gap obtained with MOSEK.

Chance-Constrained OPF in Droop-Controlled Microgrids With Power Flow Routers

High penetration of renewable generation poses challenges to power system operation due to its uncertain nature. In droop-controlled microgrids, the voltage volatility induced by renewable uncertainties is aggravated by the high droop gains. This paper proposes a chance-constrained optimal power flow (CC-OPF) problem with power flow routers (PFRs) to better regulate the voltage profile in microgrids. PFR refers to a general type of network-side controller that brings more flexibility to the power network. Comparing with the normal CC-OPF that relies on node power flexibility only, the proposed model introduces a new dimension of control from power network to enhance system performance under renewable uncertainties. Adopting a partial linearization method and an iterative algorithm allows us to address the CC-OPF problem by iteratively solving a subproblem. Since the inclusion of PFRs complicates the subproblem and makes common solvers no longer applicable directly, a semidefinite programming relaxation is used to transform each subproblem into a convex form. The proposed method is verified on a modified IEEE 33-bus system and the results show that PFRs significantly reduce the standard deviations of voltage magnitudes and contribute to mitigating the voltage volatility, which makes the system operate in a more economic and secure way.

Optimal Power Flow in AC/DC Microgrids With Enhanced Interlinking Converter Modeling

In this article, we propose an optimal power flow (OPF) paradigm for hybrid ac/dc microgrids. A meticulous model of the interlinking converter (IC) is developed and integrated into the OPF problem formulation. The resulting formulation is capable of solving the OPF of ac and dc subgrids in their standalone operations. A computationally efficient parabolic relaxation method transforms the nonconvex OPF model into a convex quadratic-constrained quadratic programming form. A sequential penalization method is applied to the relaxed OPF to achieve feasible solutions for the original formulation. The developed OPF formulation is tested on multiple modified 5-, 14-, 30-, 24-, 118-bus test systems, and a significant reduction in computational time is achieved in comparison with semidefinite programming relaxation. For a sequential penalized ac/dc OPF, the optimal values of both relaxations have a maximum difference of 0.025% for a 5-bus system. Finally, the modified IEEE 14-bus system is emulated in a real-time controller/hardware-in-the-loop setup to validate the proposed framework with continuously varying loads and IC switching status.

Learning to Solve the AC-OPF Using Sensitivity-Informed Deep Neural Networks

To shift the computational burden from real-time to offline in delay-critical power systems applications, recent works entertain the idea of using a deep neural network (DNN) to predict the solutions of the AC optimal power flow (AC-OPF) once presented load demands. As network topologies may change, training this DNN in a sample-efficient manner becomes a necessity. To improve data efficiency, this work utilizes the fact OPF data are not simple training labels, but constitute the solutions of a parametric optimization problem. We thus advocate training a sensitivity-informed DNN (SI-DNN) to match not only the OPF optimizers, but also their partial derivatives with respect to the OPF parameters (loads). It is shown that the required Jacobian matrices do exist under mild conditions, and can be readily computed from the related primal/dual solutions. The proposed SI-DNN is compatible with a broad range of OPF solvers, including a non-convex quadratically constrained quadratic program (QCQP), its semidefinite program (SDP) relaxation, and MATPOWER; while SI-DNN can be seamlessly integrated in other learning-to-OPF schemes. Numerical tests on three benchmark power systems corroborate the advanced generalization and constraint satisfaction capabilities for the OPF solutions predicted by an SI-DNN over a conventionally trained DNN, especially in low-data setups.

Small-Signal Stability-Constrained Optimal Power Flow for Inverter Dominant Autonomous Microgrids

This article details and solves a small-signal stability-constrained optimal power flow (SSSC-OPF) for inverter-based ac microgrids. To ensure a sufficient stability margin during optimal generation, a small-signal stability constraint is embedded into the conventional OPF formulation. This condition is enforced using a Lyapunov stability equation. A reduced-order model of the microgrid is adopted to alleviate the computational burden involved in solving the resulting SSSC-OPF. Even then, the resulting stability conditions are highly nonlinear and cannot be handled using the existing methods. To tackle the nonconvexity in the SSSC-OPF due to the presence of the nonlinear stability constraint, two distinct convex relaxation approaches, namely semidefinite programming and parabolic relaxations, are developed. A heuristic penalty function is added to the objective function of the relaxed SSSC-OPF, which is solved sequentially to obtain a feasible point. While off-the-shelf tools fail to produce any feasible point within hours, the proposed approach enables us to solve the SSSC-OPF in near real time. The efficacy of the proposed SSSC-OPF is evaluated by performing numerical studies on multiple benchmarks as well as real-time studies on a microgrid system built in a controller/hardware-in-the-loop setup.

BiqBin: A Parallel Branch-and-bound Solver for Binary Quadratic Problems with Linear Constraints

We present BiqBin, an exact solver for linearly constrained binary quadratic problems. Our approach is based on an exact penalty method to first efficiently transform the original problem into an instance of Max-Cut, and then to solve the Max-Cut problem by a branch-and-bound algorithm. All the main ingredients are carefully developed using new semidefinite programming relaxations obtained by strengthening the existing relaxations with a set of hypermetric inequalities, applying the bundle method as the bounding routine and using new strategies for exploring the branch-and-bound tree.Furthermore, an efficient C implementation of a sequential and a parallel branch-and-bound algorithm is presented. The latter is based on a load coordinator-worker scheme using MPI for multi-node parallelization and is evaluated on a high-performance computer.The new solver is benchmarked against BiqCrunch, GUROBI, and SCIP on four families of (linearly constrained) binary quadratic problems. Numerical results demonstrate that BiqBin is a highly competitive solver. The serial version outperforms the other three solvers on the majority of the benchmark instances. We also evaluate the parallel solver and show that it has good scaling properties. The general audience can use it as an on-line service available athttp://www.biqbin.eu.

Tight Neural Network Verification via Semidefinite Relaxations and Linear Reformulations

We present a novel semidefinite programming (SDP) relaxation that enables tight and efficient verification of neural networks. The tightness is achieved by combining SDP relaxations with valid linear cuts, constructed by using the reformulation-linearisation technique (RLT). The computational efficiency results from a layerwise SDP formulation and an iterative algorithm for incrementally adding RLT-generated linear cuts to the verification formulation. The layer RLT-SDP relaxation here presented is shown to produce the tightest SDP relaxation for ReLU neural networks available in the literature. We report experimental results based on MNIST neural networks showing that the method outperforms the state-of-the-art methods while maintaining acceptable computational overheads. For networks of approximately 10k nodes (1k, respectively), the proposed method achieved an improvement in the ratio of certified robustness cases from 0% to 82% (from 35% to 70%, respectively).

Optimal H2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization

In this paper, we compute a (local) optimal reduced order model that matches a prescribed set of moments of a stable linear time-invariant system of high dimension. We fix the interpolation points and parametrize the models achieving moment-matching in a set of free parameters. Based on the parametrization and using the H2-norm of the approximation error as the objective function, we derive a nonconvex optimization problem, i.e., we search for the optimal free parameters to determine the model yielding the minimal H2-norm of the approximation error. Furthermore, we provide the necessary first-order optimality conditions in terms of the controllability and the observability Gramians of a minimal realization of the error system. We then propose two gradient-type algorithms to compute the (local) optimal models, with mathematical guarantees on the convergence. We also derive convex semidefinite programming relaxations for the nonconvex Problem, under the assumption that the error system admits block-diagonal Gramians, and derive sufficient conditions to guarantee the block diagonalization. The solutions resulting at each step of the proposed algorithms guarantee the achievement of the imposed moment matching conditions. The second gradient-based algorithm exhibits the additional property that, when stopped, yields a stable approximation with a reduced H2-error norm. We illustrate the theory on a CD-player and on a discretized heat equation.

Necessary and Sufficient Conditions for Rank-One-Generated Cones

A closed convex conic subset [Formula: see text] of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all of its extreme rays are generated by rank-one matrices. The ROG property of [Formula: see text] is closely related to the exactness of semidefinite program (SDP) relaxations of nonconvex quadratically constrained quadratic programs (QCQPs) related to [Formula: see text]. We consider the case where [Formula: see text] is obtained as the intersection of the PSD cone with finitely many homogeneous linear matrix inequalities and conic constraints and identify sufficient conditions that guarantee that [Formula: see text] is ROG. Our general framework allows us to recover a number of well-known results from the literature. In the case of two linear matrix inequalities, we also establish the necessity of our sufficient conditions. This extends one of the few settings from the literature—the case of one linear matrix inequality and the S-lemma—where an explicit characterization for the ROG property exists. Finally, we show how our ROG results on cones can be translated into inhomogeneous SDP exactness results and convex hull descriptions in the original space of a QCQP. We close with a few applications of these results; specifically, we recover the well-known perspective reformulation of a simple mixed-binary set via the ROG toolkit. Funding: This work was supported by the Office of Naval Research [Grant N00014-19-1-2321] with the National Science Foundation (NSF) Division of Civil, Mechanical and Manufacturing Innovation [Grant CMMI-1454548] and Frank A. and Helen E. Risch Faculty Development Chair. Part of this work was done while the second author was visiting the Simons Institute for the Theory of Computing, which was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through the NSF Division of Computing and Communication Foundations [Grant CCF-1740425].

Conic optimization method for voltage stability assessment of active distribution networks with different wind turbine models

This paper presents a conic optimization method based on Semi-Definite Programming (SDP) relaxation for calculating the Static Voltage Stability Limitation (SVSL) in Active Distribution Networks (ADNs), considering correlated uncertainties associated with wind speeds and load forecasting. This method analyses the effect of wind generation on the SVSL interval evaluation with consideration of different wind turbine models. With the proposed Interval Ellipsoid Hybrid Convex Model (IEHCM), a conic optimization solution is presented, which may yield better results in dealing with correlated uncertainties and maintain high accuracy and convergence stability as compared to other alternative methods. The proposed method is tested on a modified IEEE 33-bus systems and several IEEE standard systems to show its superiority.

On computational capabilities of Ising machines based on nonlinear oscillators

Dynamical Ising machines are actively investigated from the perspective of finding efficient heuristics for NP-hard optimization problems. However, the existing data demonstrate super-polynomial scaling of the running time with the system size, which is incompatible with large NP-hard problems. We show that oscillator networks implementing the Kuramoto model of synchronization are capable of demonstrating polynomial scaling. The dynamics of these networks is related to the semidefinite programming relaxation of the Ising model ground state problem. Consequently, such networks, as we numerically demonstrate, are capable of producing the best possible approximation in polynomial time. To reach such performance, however, the reconstruction of the binary Ising state (rounding) must be specially addressed. We demonstrate that commonly implemented forced collapse to a close-to-Ising state may diminish the computational capabilities up to their complete invalidation. Therefore, consistent treatment of rounding may cardinally improve various operation metrics of already existing and upcoming dynamical Ising machines.

Multi-terminal phase-changing soft open point SDP modeling for imbalance mitigation in active distribution networks

Active distribution networks (ADNs) are capable of mitigating phase imbalance caused by various operational conditions, including uneven growth of single-phase and intermittent distributed energy resources (DERs), incurring financial losses or costly infrastructure reinforcements. In this paper, the research gap for a flexible phase imbalance mitigating solution is addressed by proposing a multi-terminal phase-changing soft open point (PC-SOP). It is explored in detail on balancing the power flows and compared with other different types and ways of connection (including two-terminal and conventional). Then operational strategies based on different cases are presented for imbalance mitigation. Semidefinite programming (SDP) relaxation is utilized to convert the original non-convex nonlinear model into an SDP model which can be solved efficiently by commercial solvers. Two case studies demonstrations are conducted on IEEE 13-node and 123-node three-phase networks. It is found that multi-terminal PC-SOPs can minimize power losses by between 5.56 % and 28.98% and have better voltage control (all buses operate in the allowed voltage range [0.94, 1.10]) and less PV curtailment (reduced by at least 6.31 MW/24 h and 0.63 MW/24 h for the two test networks separately) when compared to conventional SOP technologies.

Resource Allocation for Intelligent Reflecting Surface Assisted Wireless Powered IoT Systems With Power Splitting

This paper proposes a new transmission policy for intelligent reflecting surface (IRS) empowered wireless powered internet of things systems. Particularly, an energy station (ES) wirelessly charges for multiple IoT devices during downlink wireless energy transfer (WET) and then these devices deliver their own message to an access point (AP) during uplink wireless information transfer (WIT). Also, an IRS is deployed to improve energy harvesting and data transmission capabilities. To enhance self-sustainability of the IRS, the IRS harvests energy from the ES based on the harvest-then-transmit protocol. In this paper, we maximize the sum throughput via optimizing the phase shifts of the IRS, the transfer time scheduling as well as the power splitting ratio. Due to the non-convexity of the formulated problem, we divide the problem into two sub-problems, each of which can be handled separately. Then, we adopt an alternating optimization (AO) algorithm with the semidefinite programming (SDP) relaxation. Also, we consider a special case where the circuit power consumption of IoT devices can be neglected. In this case, we derive a closed form solution for the optimal transmission time slots, power allocation and phase shift by the Lagrange dual method. Finally, numerical evaluations validate effectiveness of the proposed scheme, which significantly benefits from the IRS in improving network throughput.

Hybrid quantum-classical algorithms for approximate graph coloring

We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX-k-CUT, the problem of finding an approximate k-vertex coloring of a graph. We compare this proposal to the best known classical and hybrid classical-quantum algorithms. First, we show that the standard (non-recursive) QAOA fails to solve this optimization problem for most regular bipartite graphs at any constant level p: the approximation ratio achieved by QAOA is hardly better than assigning colors to vertices at random. Second, we construct an efficient classical simulation algorithm which simulates level-1 QAOA and level-1 RQAOA for arbitrary graphs. In particular, these hybrid algorithms give rise to efficient classical algorithms, and no benefit arising from the use of quantum mechanics is to be expected. Nevertheless, they provide a suitable testbed for assessing the potential benefit of hybrid algorithm: We use the simulation algorithm to perform large-scale simulation of level-1 QAOA and RQAOA with up to 300 qutrits applied to ensembles of randomly generated 3-colorable constant-degree graphs. We find that level-1 RQAOA is surprisingly competitive: for the ensembles considered, its approximation ratios are often higher than those achieved by the best known generic classical algorithm based on rounding an SDP relaxation. This suggests the intriguing possibility that higher-level RQAOA may be a potentially useful algorithm for NISQ devices.

Hidden Integrality and Semirandom Robustness of SDP Relaxation for Sub-Gaussian Mixture Model

We consider the problem of estimating the discrete clustering structures under the sub-Gaussian mixture model. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solution to the SDP is not integer-valued in general, its estimation error can be upper bounded by that of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. In addition, we show that the SDP relaxation is robust under the semirandom setting in which an adversary can modify the data generated from the mixture model. In particular, we generalize the hidden integrality property to the semirandom model and thereby show that SDP achieves the optimal error bound in this setting. These results together highlight the “global-to-local” mechanism that drives the performance of the SDP relaxation. To the best of our knowledge, our result is the first exponentially decaying error bound for convex relaxations of mixture models. A corollary of our results shows that in certain regimes, the SDP solutions are in fact integral and exact. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of [Formula: see text] fraction of the points for any [Formula: see text]. As a special case, we show that under the [Formula: see text]-dimensional stochastic ball model, SDP achieves nontrivial (sometimes exact) recovery when the center separation is as small as [Formula: see text], which improves upon previous exact recovery results that require constant separation.