Semidefinite Programming Relaxation Research Articles

An SDP relaxation in the complex domain for the efficient coordination of BESS and DGs in single-phase distribution grids while considering reactive power capabilities

This research proposes an efficient energy management system design for medium-voltage distribution networks from a perspective of convex optimization. The exact nonlinear programming problem that represents the operation of distributed energy resources (DERs), such as batteries and photovoltaic sources, is approximated as an equivalent semi-definite programming (SDP) problem. This SDP approximation is achieved in the complex domain using Hermitian matrices, which are the equivalent of positive semi-definite matrices in the real number domain. Two test feeders, one with 27 nodes and the other with 33 nodes, are modified to match the characteristics of Colombian rural and urban networks. The 27-bus grid is adapted to accommodate the average demand and solar generation conditions of the municipality of Capurganá, while those of the city of Medellín are used for the 33-bus grid. One of the main contributions of this research is that it considers a DER operation with variable power factors, which allows for significant improvements compared to the traditional use of a unitary power factor scheme. Numerical comparisons are conducted using metaheuristic optimizers, including parallel versions of the particle swarm optimizer, the vortex search algorithm, and the continuous version of the genetic algorithm. These comparisons demonstrate the effectiveness of the proposed SDP approximation. All simulations are carried out in the MATLAB software (version 2022b), using the disciplined convex optimizer (CVX) and the SDPT3 and SEDUMI solvers.

Network partition and distributed voltage coordination control strategy of active distribution network system considering photovoltaic uncertainty

Aiming at the voltage exceeding problem caused by the introduction of Photovoltaic (PV) in Active distribution network (ADN), a comprehensive electrical distance index was established considering active and reactive voltage sensitivity, and the uncertainty of traditional PV probability density function (PDF) was optimized by random simulation method. The electrical distance expectation matrix between nodes was established by discretization of PV output probability distribution characteristics, and the ADN system was partitioned by affinity propagation (AP) algorithm. Taking active power loss as the objective function, the non-convex reactive power optimization control problem is transformed into a convex quadratic programming problem by LinDistFlow equation. An improved alternating direction method of multipliers (ADMM) which adaptively adjusts the penalty parameters is used to convert the finite boundaries of adjacent regions into data. By coordinating on-load tap-changer (OLTC), capacitor banks (CBs), and PV inverters on different time scales, the fast optimal control of global voltage in ADN is realized. The proposed method is tested on IEEE 33-bus and IEEE 123-bus distribution systems. After the optimization control algorithm described in this paper, the system network loss was reduced by 3.88%. Compared with the semi-definite programming relaxation, the calculation speed of LinDistFlow equation is increased by 72.2%. The results fully verify the feasibility and high efficiency of the control strategy described in this paper.

Tight Sum-of-squares Lower Bounds for Binary Polynomial Optimization Problems

For binary polynomial optimization problems of degree 2d with n variables Sakaue, Takeda, Kim, and Ito [ 33 ] proved that the \(\lceil \frac{n+2d-1}{2}\rceil\) th semidefinite (SDP) relaxation in the SoS/Lasserre hierarchy of SDP relaxations provides the exact optimal value. When n is an odd number, we show that their analysis is tight, i.e., we prove that \(\frac{n+2d-1}{2}\) levels of the SoS/Lasserre hierarchy are also necessary. Laurent [ 24 ] showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull of a linear representation of a set with no integral points. She conjectured that the SoS/Lasserre rank for the same problem is n -1. In this article, we disprove this conjecture and derive lower and upper bounds for the rank.

On semidefinite descriptions for convex hulls of quadratic programs

Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard semidefinite program (SDP) relaxation. In this paper we study when the convex hull of the epigraph of a QCQP coincides with the projected epigraph of the SDP relaxation. We present a sufficient condition for convex hull exactness and show that this condition is further necessary under an additional geometric assumption. The sufficient condition is based on geometric properties of Γ, the cone of convex Lagrange multipliers, and its relatives Γ1 and Γ∘.

Entropy constraints for ground energy optimization

We study the use of von Neumann entropy constraints for obtaining lower bounds on the ground energy of quantum many-body systems. Known methods for obtaining certificates on the ground energy typically use consistency of local observables and are expressed as semidefinite programming relaxations. The local marginals defined by such a relaxation do not necessarily satisfy entropy inequalities that follow from the existence of a global state. Here, we propose to add such entropy constraints that lead to tighter convex relaxations for the ground energy problem. We give analytical and numerical results illustrating the advantages of such entropy constraints. We also show limitations of the entropy constraints we construct: they are implied by doubling the number of sites in the relaxation and as a result they can at best lead to a quadratic improvement in terms of the matrix sizes of the variables. We explain the relation to a method for approximating the free energy known as the Markov Entropy Decomposition method.

Learning Guarantee of SDP Relaxation on Directed Stochastic Block Models

Community detection in networks has been a focal point in various scientific domains, but the study of directed networks remains relatively under-explored despite their prevalence and importance in capturing real-world systems. This work addresses this research gap by focusing on Directed Stochastic Block Models (DSBMs), a natural extension of traditional Stochastic Block Models (SBMs) to directed graphs. The inherent complexity of directionality in DSBMs makes them challenging to analyze, requiring new mathematical frameworks and computational approaches. We introduce an augmented matrix to encapsulate the directional relationships within these networks, providing a nuanced perspective for further analysis. In this work, we prove the information-theoretical threshold for exact recovery in the DSBMs and propose an SDP relaxation that can achieve this threshold, thereby contributing to the theoretical understanding of community detection in the realm of directed graphs.

Solving Sparse Separable Bilinear Programs Using Lifted Bilinear Cover Inequalities

Recently a class of second-order cone representable convex inequalities called lifted bilinear cover inequalities were introduced, which are valid for a set described by a separable bilinear constraint together with bounds on variables. In this paper, we study the computational potential of these inequalities for separable bilinear optimization problems. We first prove that the semidefinite programming relaxation provides no benefit over the McCormick relaxation for such problems. We then design a simple randomized separation heuristic for lifted bilinear cover inequalities. In our computational experiments, we separate many rounds of these inequalities starting from McCormick’s relaxation of instances where each constraint is a separable bilinear constraint set. We demonstrate that there is a significant improvement in the performance of a state-of-the-art global solver in terms of gap closed, when these inequalities are added at the root node compared with when they are not. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: S. S. Dey gratefully acknowledges the support by the Office of Naval Research [Grant N000141912323].

On semidefinite programming relaxations for a class of robust SOS-convex polynomial optimization problems

On semidefinite programming relaxations for a class of robust SOS-convex polynomial optimization problems

A Unified Framework for Tuning Hyperparameters in Clustering Problems

Selecting hyperparameters for unsupervised learning problems is challenging in general due to the lack of ground truth for validation. Despite the prevalence of this issue in statistics and machine learning, especially in clustering problems, there are not many methods for tuning these hyperparameters with theoretical guarantees. In this paper, we provide a framework with provable guarantees for selecting hyperparameters in a number of distinct models. We consider both the subgaussian mixture model and network models to serve as examples of i.i.d. and non-i.i.d. data. We demonstrate that the same framework can be used to choose the Lagrange multipliers of penalty terms in semi-definite programming (SDP) relaxations for community detection, and the bandwidth parameter for constructing kernel similarity matrices for spectral clustering. By incorporating a cross-validation procedure, we show the framework can also do consistent model selection for network models. Using a variety of simulated and real data examples, we show that our framework outperforms other widely used tuning procedures in a broad range of parameter settings.

Pricing Economic Dispatch With AC Power Flow Via Local Multipliers and Conic Relaxation

We analyze two locational marginal pricing schemes in electricity markets derived from an economic dispatch (ED) problem with AC power flow equations that define a non-convex feasible set. The first among these prices, termed AC-LMPs, are derived from Lagrange multipliers that satisfy Karush-Kuhn-Tucker conditions for stationarity/local optimality of the non-convex ED problem. The second, called SDP-LMPs, are derived from optimal dual multipliers of the semidefinite programming (SDP)-based convex relaxation of the ED problem. We establish that AC-LMPs and SDP-LMPs are derived from Lagrange dual-equivalent problems. Hence, they coincide under zero duality gap, but may not be equal when the gap is non-zero or the AC-LMPs are associated with a stationary/locally optimal, but not globally optimal, dispatch. SDP-LMPs share interesting parallels with convex hull prices (CHPs), being derived from the Lagrangian dual of the non-convex ED problem. However, there are important differences. For example, the epigraphs of the SDP relaxation of the ED problem and the latter, parameterized by the nodal power demands, may not enjoy the relationship that CHPs satisfy and derive their name from. Also, while CHPs minimize a form of side-payments, SDP-LMPs may not. We prove that AC-LMPs, (and SDP-LMPs under zero duality gap) guarantee revenue adequacy under a condition that is sufficient, but not necessary. Finally, the SDP-LMPs are shown to equal SOCP-DLMPs, that are distribution locational marginal prices derived with second-order cone programming-based relaxations of power flow equations over radial distribution networks. We illustrate our theoretical findings through numerical experiments.

Enhancing resilience by outage management strategy based on semidefinite programming relaxation method in unbalanced microgrids

During electrical power outages caused by extreme events, microgrid operators (MGOs) attempt to restore as much electrical load as possible. This work suggests a strategy for outage management (OM) to improve microgrid resilience by using two optimal actions: distribution feeder reconfiguration (DFR) and scheduling of distributed energy resources (DERs). After a line fault, the radial network topology is determined by the proposed model using the rank of the incidence matrix. The proposed model is formulated by an analytical optimization method created by semidefinite programming (SDP) relaxation. The SDP changes the non-convex and nonlinear model suggested for OM into an approximated convex model, which commercial software applies. The proposed model considers the uncertain behavior of non-dispatchable DERs and the electrical demands that deal with an information gap decision theory (IGDT) based on a risk-averse strategy. To improve the proposed model's flexibility, the shortened electrical demand as a demand response program (DRP) is considered. The aim of optimization is the minimization of the accumulative cost for dispatchable DER operation and load reduction. The suggested SDP model is figured out using the MOSEK solver in GAMS software. Using the offered model in the 69-bus unbalanced test system displays that the SDP model averagely decreases total operation cost and execution time by 1.04% and 61.29% on all scenarios in comparison with the conventional GAMS model.

Intelligent Reflecting Surface-Assisted Wireless Powered Heterogeneous Networks

In this paper, we introduce an intelligent reflecting surface (IRS)-assisted wireless powered heterogeneous network (WPHN) consisting of two heterogeneous groups of devices. Specifically, one group of devices, i.e., energy-harvesting devices (EHDs), are charged by external energy supplies, while the other group of devices, i.e., non-energy-harvesting devices (NEHDs), are powered by internal energy supplies. An IRS aims to participate in the wireless energy transfer (WET) in downlink and the wireless information transfer (WIT) in the uplink. A sum throughput maximization problem is formulated subject to the constraints of individual energy consumption, transmission time scheduling, and IRS phase shifts. To cope with the non-convexity of the problem, we first derive the optimal IRS phase shifts of the uplink WIT independently. Next, the semi-definite programming (SDP) relaxation is adopted to recast this non-convex problem into the convex one, which can be numerically solved. Then, a novel low-complexity scheme is developed to gain more insights and mitigate the computational complexity induced by the SDP relaxation. In particular, the dual problem and Karush-Kuhn-Tucker conditions are first utilized to obtain the optimal transmission time scheduling. Then, we propose a method based on Riemannian manifold optimization to compute the optimal IRS phase shifts of the downlink WET in closed-form. Finally, simulation results are presented to verify the optimality of our proposed scheme, and highlight the benefits induced by the IRS to coordinate these heterogeneous devices.

An Improved Approximation Algorithm for Quantum Max-Cut on Triangle-Free Graphs

We give an approximation algorithm for Quantum Max-Cut which works by rounding an SDP relaxation to an entangled quantum state. The SDP is used to choose the parameters of a variational quantum circuit. The entangled state is then represented as the quantum circuit applied to a product state. It achieves an approximation ratio of 0.582 on triangle-free graphs. The previous best algorithms of Anshu, Gosset, Morenz, and Parekh, Thompson achieved approximation ratios of 0.531 and 0.533 respectively. In addition, we study the EPR Hamiltonian, which we argue is a natural intermediate problem which isolates some key quantum features of local Hamiltonian problems. For the EPR Hamiltonian, we give an approximation algorithm with approximation ratio 1/2 on all graphs.

Global optimization for cardinality-constrained minimum sum-of-squares clustering via semidefinite programming

AbstractThe minimum sum-of-squares clustering (MSSC), or k-means type clustering, has been recently extended to exploit prior knowledge on the cardinality of each cluster. Such knowledge is used to increase performance as well as solution quality. In this paper, we propose a global optimization approach based on the branch-and-cut technique to solve the cardinality-constrained MSSC. For the lower bound routine, we use the semidefinite programming (SDP) relaxation recently proposed by Rujeerapaiboon et al. (SIAM J Optim 29(2):1211–1239, 2019). However, this relaxation can be used in a branch-and-cut method only for small-size instances. Therefore, we derive a new SDP relaxation that scales better with the instance size and the number of clusters. In both cases, we strengthen the bound by adding polyhedral cuts. Benefiting from a tailored branching strategy which enforces pairwise constraints, we reduce the complexity of the problems arising in the children nodes. For the upper bound, instead, we present a local search procedure that exploits the solution of the SDP relaxation solved at each node. Computational results show that the proposed algorithm globally solves, for the first time, real-world instances of size 10 times larger than those solved by state-of-the-art exact methods.

An accelerated primal‐dual method for semi‐definite programming relaxation of optimal power flow

AbstractThe application of a semi‐definite programming (SDP) approach to the Alternating Current Optimal Power Flow problem has attracted significant attention in recent years. However, the SDP relaxation of optimal power flow (OPF) can be computationally intensive and lead to memory issues when dealing with large‐scale power systems. To overcome these challenges, we have developed APD–SDP, an optimisation solver based on a first‐order primal–dual algorithm. This framework incorporates various acceleration techniques, such as rescaling, step size decay and reset, adaptive line search, and restart, to improve efficiency. To further speed up computations, we have developed a customised eigenvalue decomposition component by exploiting the 3 × 3 block structure in the dual SDP formulation. Experimental results demonstrate that APD–SDP outperforms other commercial and open‐source SDP solvers on large‐scale and high‐dimensional PGLib‐OPF datasets.

Near-optimal bounds for generalized orthogonal Procrustes problem via generalized power method

Given multiple point clouds, how to find the rigid transform (rotation, reflection, and shifting) such that these point clouds are well aligned? This problem, known as the generalized orthogonal Procrustes problem (GOPP), has found numerous applications in statistics, computer vision, and imaging science. While one commonly-used method is finding the least squares estimator, it is generally an NP-hard problem to obtain the least squares estimator exactly due to the notorious nonconvexity. In this work, we apply the semidefinite programming (SDP) relaxation and the generalized power method to solve this generalized orthogonal Procrustes problem. In particular, we assume the data are generated from a signal-plus-noise model: each observed point cloud is a noisy copy of the same unknown point cloud transformed by an unknown orthogonal matrix and also corrupted by additive Gaussian noise. We show that the generalized power method (equivalently alternating minimization algorithm) with spectral initialization converges to the unique global optimum to the SDP relaxation, provided that the signal-to-noise ratio is high. Moreover, this limiting point is exactly the least squares estimator and also the maximum likelihood estimator. Our theoretical bound is near-optimal in terms of the information-theoretic limit (only loose by a factor of the dimension and a log factor). Our results significantly improve the state-of-the-art results on the tightness of the SDP relaxation for the generalized orthogonal Procrustes problem, an open problem posed by Bandeira et al. (2014) [8].

Relative Transformation Estimation Based on Fusion of Odometry and UWB Ranging Data

In this article, we study the problem of estimating the four-degree-of-freedom (3-D position and heading) robot-to-robot relative frame transformation using onboard odometry and interrobot distance measurements. First, we present a theoretical analysis of the problem, namely, the derivation and interpretation of the Cramèr–Rao lower bound, the Fisher information matrix, and its determinant. Second, we propose optimization-based solutions, including a quadratically constrained quadratic programming (QCQP) formulation and its semidefinite programming (SDP) relaxation. Third, based on the theoretical results, we can detect singular configurations as well as measure the uncertainty of each individual parameter. We perform extensive simulations and real-life experiments with aerial robots to show that the proposed QCQP and SDP methods can outperform state-of-the-art approaches, especially in geometrically poor or large measurement noise conditions. In general, the QCQP method provides the best results at the expense of computational time, while the SDP method runs much faster and is sufficiently accurate in most cases.

Facial reduction for the Shor SDP relaxation of QCQPs

Facial reduction for the Shor SDP relaxation of QCQPs

A Semidefinite Relaxation Based Branch-and-Bound Method for Tight Neural Network Verification

We introduce a novel method based on semidefinite program (SDP) for the tight and efficient verification of neural networks. The proposed SDP relaxation advances the present state of the art in SDP-based neural network verification by adding a set of linear constraints based on eigenvectors. We extend this novel SDP relaxation by combining it with a branch-and-bound method that can provably close the relaxation gap up to zero. We show formally that the proposed approach leads to a provably tighter solution than the present state of the art. We report experimental results showing that the proposed method outperforms baselines in terms of verified accuracy while retaining an acceptable computational overhead.

Bounding real tensor optimizations via the numerical range

A new method of using the numerical range of a matrix to bound the optimal value of certain optimization problems over real tensor product vectors is presented. This bound is stronger than the trivial bounds based on eigenvalues and can be computed significantly faster than bounds provided by semidefinite programming relaxations. Numerous applications to other hard linear algebra problems are discussed, such as showing that a real subspace of matrices contains no rank-one matrix, and showing that a linear map acting on matrices is positive.