This paper investigates the impact of the changes in the demand of power systems on the quality of the solution procured by the convex relaxation methods for the AC optimal power flow (ACOPF) problem. This investigation needs various measures to evaluate the tightness of the solution procured by the convex relaxation approaches. Therefore, three tightness measures are leveraged to illustrate the performance of convex relaxation methods under different demand scenarios. The main issue of convex relaxation methods is recovering an optimal solution which is not necessarily feasible for the original non-convex problem in networks with cycles. Thus, a cycle measure is introduced to evaluate the performance of relaxation schemes. The presented case study investigates the merit of using various tightness measures to evaluate the performance of various relaxation methods under different circumstances.
Optimal power flow (OPF) is the fundamental mathematical model in power system operations. Improving the solution quality of OPF provide huge economic and engineering benefits. The convex reformulation of the original nonconvex alternating current OPF (ACOPF) model gives an efficient way to find the global optimal solution of ACOPF but suffers from the relaxation gaps. The existence of relaxation gaps hinders the practical application of convex OPF due to the AC-infeasibility problem. We evaluate and improve the tightness of the convex ACOPF model in this paper. Various power networks and noda…
This paper presents a reformulation for the automatic generation control (AGC) in a decomposed convex relaxation algorithm. It finds an optimal solution to the AC optimal power flow (ACOPF) problem that is secure against a large set of contingencies. The original ACOPF problem which represents the system without contingency constraints, is convexified by applying the second-order cone relaxation method. The contingencies are filtered to distinguish those that will be treated with preventive actions from those that will be left for corrective actions. The selected contingencies for preventive a…
We derive the branch ampacity constraint associated to power losses for the convex optimal power flow (OPF) model based on the branch flow formulation. The branch ampacity constraint derivation is motivated by the physical interpretation of the transmission line Π-model and practical engineering considerations. We rigorously prove and derive: (i) the loop constraint of voltage phase angle, required to make the branch flow model valid for meshed power networks, is a relaxation of the original nonconvex alternating current optimal power flow (o-ACOPF) model; (ii) the necessary conditions to reco…
The optimal reactive power dispatch (ORPD) problem is an alternating current optimal power flow (ACOPF) problem where discrete control devices for regulating the reactive power, such as shunt elements and tap changers, are considered. The ORPD problem is modelled as a mixed-integer nonlinear optimization problem and its complexity is increased compared to the ACOPF problem, which is highly nonconvex and generally hard to solve. Recently, convex relaxations of the ACOPF problem have attracted a significant interest since they can lead to global optimality. We propose a tight conic relaxation of…
The alternating-current optimal power flow (ACOPF) is one of the best known non-convex non-linear optimisation problems. We present a novel re-formulation of ACOPF, which is based on lifting the rectangular power-voltage rank-constrained formulation, and makes it possible to derive alternative SDP relaxations. For those, we develop a first-order method based on the parallel coordinate descent with a novel closed-form step based on roots of cubic polynomials.
We consider a robust optimization problem in an electric power system under uncertain demand and availability of renewable energy resources. Solving the deterministic alternating current optimal power flow (ACOPF) problem has been considered challenging since the 1960s due to its nonconvexity. Linear approximation of the AC power flow system sees pervasive use, but does not guarantee a physically feasible system configuration. In recent years, various convex relaxation schemes for the ACOPF problem have been investigated, and under some assumptions, a physically feasible solution can be recove…
We have observed an interesting, yet unexplained, phenomenon: Semidefinite programming (SDP) based relaxations of maximum likelihood estimators (MLE) tend to be tight in recovery problems with noisy data, even when MLE cannot exactly recover the ground truth. Several results establish tightness of SDP based relaxations in the regime where exact recovery from MLE is possible. However, to the best of our knowledge, their tightness is not understood beyond this regime. As an illustrative example, we focus on the generalized Procrustes problem.
Transmission network expansion planning is a mixed-integer optimization problem, whose solution is used to guide future investment in transmission equipment. An approach is presented to find the global solution of the transmission planning problem using an AC network model. The approach builds on the semidefinite relaxation of the AC optimal power flow problem (ACOPF); its computational engine is a new specialized branch-and-cut algorithm for transmission expansion planning to deal with the underlying mixed-integer ACOPF problem. Valid inequalities that are based on specific knowledge of the e…
Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the…
This paper studies an optimization problem on the sum of traces of matrix quadratic forms on $m$ orthogonal matrices, which can be considered as a generalization of the synchronization of rotations. While the problem is nonconvex, the paper shows that its semidefinite programming relaxation can solve the original nonconvex problems exactly, under an additive noise model with small noise in the order of $O(-m^{1/4})$, where $m$ is the number of orthogonal matrices. This result can be considered as a generalization of existing results on phase synchronization.
Maximum likelihood estimation problems are, in general, intractable optimization problems. As a result, it is common to approximate the maximum likelihood estimator (MLE) using convex relaxations. In some cases, the relaxation is tight: it recovers the true MLE. Most tightness proofs only apply to situations where the MLE exactly recovers a planted solution (known to the analyst). It is then sufficient to establish that the optimality conditions hold at the planted signal. In this paper, we study an estimation problem (angular synchronization) for which the MLE is not a simple function of the …
This paper studies an optimization problem on the sum of traces of matrix quadratic forms in $m$ semi-orthogonal matrices, which can be considered as a generalization of the synchronization of rotations. While the problem is nonconvex, the paper shows that its semidefinite programming relaxation solves the original nonconvex problems exactly with high probability, under an additive noise model with small noise in the order of $O(m^{1/4})$. In addition, it shows that the solution of a nonconvex algorithm considered in Won, Zhou, and Lange [SIAM J. Matrix Anal. Appl., 2 (2021), pp. 859-882] is a…
In this paper, we consider a fundamental problem in modern digital communications known as multi-input multi-output (MIMO) detection, which can be formulated as a complex quadratic programming problem subject to unit-modulus and discrete argument constraints. Various semidefinite relaxation (SDR) based algorithms have been proposed to solve the problem in the literature. In this paper, we first show that the conventional SDR is generally not tight for the problem. Then, we propose a new and enhanced SDR and show its tightness under an easily checkable condition, which essentially requires the …
Many steady-state problems in power systems, including rectangular power-voltage formulations of optimal power flows in the alternating-current model (ACOPF), can be cast as polynomial optimisation problems (POP). For a POP, one can derive strong convex relaxations, or rather hierarchies of ever stronger, but ever larger relaxations. We study means of switching from solving the convex relaxation to Newton method working on a non-convex Lagrangian of the POP.
Max-Min Bilinear Completely Positive Programs: A Semidefinite Relaxation with Tightness Guarantees
Lee Kong Chian School of Business, Singapore Management University（新加坡管理大学李光前商学院）；Department of Mathematics, Hong Kong Baptist University（香港浸会大学数学系）；Institute for Research and Continuing Education, Hong Kong Baptist University（香港浸会大学持续教育学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
In this paper, we show that the standard semidefinite programming (SDP) relaxation of altering current optimal power flow (AC OPF) can be equivalently reformulated as second-order cone programming (SOCP) relaxation with maximal clique- and cycle-based SDP feasibility constraints. The formulation is based on the positive semi-definite (PSD) matrix completion theorem, which states that if all sub-matrices corresponding to maximal cliques in a chordal graph are PSD, then the partial matrix related to the chordal graph can be completed as a full PSD matrix. Existing methods in [1] first construct …
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
The robustness of a neural network to adversarial examples can be provably certified by solving a convex relaxation. If the relaxation is loose, however, then the resulting certificate can be too conservative to be practically useful. Recently, a less conservative robustness certificate was proposed, based on a semidefinite programming (SDP) relaxation of the ReLU activation function. In this paper, we describe a geometric technique that determines whether this SDP certificate is exact, meaning whether it provides both a lower-bound on the size of the smallest adversarial perturbation, as well…
The rotation search problem aims to find a 3D rotation that best aligns a given number of point pairs. To induce robustness against outliers for rotation search, prior work considers truncated least-squares (TLS), which is a non-convex optimization problem, and its semidefinite relaxation (SDR) as a tractable alternative. Whether this SDR is theoretically tight in the presence of noise, outliers, or both has remained largely unexplored. We derive conditions that characterize the tightness of this SDR, showing that the tightness depends on the noise level, the truncation parameters of TLS, and …
Group synchronization aims to recover the group elements from their noisy pairwise measurements. It has found many applications in community detection, clock synchronization, and joint alignment problem. This paper focuses on the orthogonal group synchronization which is often used in cryo-EM and computer vision. However, it is generally NP-hard to retrieve the group elements by finding the least squares estimator. In this work, we first study the semidefinite programming (SDP) relaxation of the orthogonal group synchronization and its tightness, i.e., the SDP estimator is exactly equal to the…
On Semidefinite Relaxations for Matrix-Weighted State-Estimation Problems in Robotics
University of Toronto（多伦多大学）；University of Toronto Robotics Institute（多伦多大学机器人研究所）；Inria（法国国家信息与自动化研究所）；École Normale Supérieure（巴黎高等师范学院）；PSL University（巴黎文理大学）
The multiple-input multiple-output (MIMO) detection problem, a fundamental problem in modern digital communications, is to detect a vector of transmitted symbols from the noisy outputs of a fading MIMO channel. The maximum likelihood detector can be formulated as a complex least-squares problem with discrete variables, which is NP-hard in general. Various semidefinite relaxation (SDR) methods have been proposed in the literature to solve the problem due to their polynomial-time worst-case complexity and good detection error rate performance. In this paper, we consider two popular classes of SD…
Consider a polynomial optimisation problem, whose instances vary continuously over time. We propose to use a coordinate-descent algorithm for solving such time-varying optimisation problems. In particular, we focus on relaxations of transmission-constrained problems in power systems. On the example of the alternating-current optimal power flows (ACOPF), we bound the difference between the current approximate optimal cost generated by our algorithm and the optimal cost for a relaxation using the most recent data from above by a function of the properties of the instance and the rate of change t…
Many modern applications involve the acquisition of noisy modulo samples of a function $f$, with the goal being to recover estimates of the original samples of $f$. For a Lipschitz function $f:[0,1]^d \to \mathbb{R}$, suppose we are given the samples $y_i = (f(x_i) + η_i)\bmod 1; \quad i=1,\dots,n$ where $η_i$ denotes noise. Assuming $η_i$ are zero-mean i.i.d Gaussian's, and $x_i$'s form a uniform grid, we derive a two-stage algorithm that recovers estimates of the samples $f(x_i)$ with a uniform error rate $O((\frac{\log n}{n})^{\frac{1}{d+2}})$ holding with high probability. The first stage …
Optimal power flow (OPF) is an important problem in the operation of electric power systems. Due to the OPF problem's non-convexity, there may exist multiple local optima. Certifiably obtaining the global solution is important for certain applications of OPF problems. Many global optimization techniques compute an optimality gap that compares the achievable objective value corresponding to the feasible point from a local solution algorithm with the objective value bound from a convex relaxation technique. Rather than the traditional practice of completely separating the local solution and conv…
Regularization vs. Relaxation: A conic optimization perspective of statistical variable selection
Department of Mathematics, Washington State University（数学系，华盛顿州立大学）；Department of Statistics, University of Connecticut（统计系，康涅狄格大学）；Department of Industrial and Systems Engineering, University of Wisconsin-Madison（工业与系统工程系，威斯康星大学麦迪逊分校）
The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a large class of random Laplacian matrices, this bound is essentially tight: the largest eigenvalue is, up to lower order terms, often the size of the largest diagonal entry. Besides being a simple tool to obtain precise estimates on the largest eigenvalue of a large class of random Laplacian matrices, our main result settles a number of open problems related to the tightness of certain convex relaxation-based algorithms. It easily implies the optimality of the semidefinite relaxation…
Optimal power flow (OPF) is a key problem in power system operations. OPF problems that use the nonlinear AC power flow equations to accurately model the network physics have inherent challenges associated with non-convexity. To address these challenges, recent research has applied various convex relaxation approaches to OPF problems. The QC relaxation is a promising approach that convexifies the trigonometric and product terms in the OPF problem by enclosing these terms in convex envelopes. The accuracy of the QC relaxation strongly depends on the tightness of these envelopes. This paper pres…
This paper focuses on the AC Optimal Power Flow (OPF) problem for multi-phase systems. Particular emphasis is given to systems with high integration of renewables, where adjustments of the real and reactive output powers from renewable sources of energy are necessary in order to enforce voltage regulation. The AC OPF problem is known to be nonconvex (and, in fact, NP-hard). Convex relaxation techniques have been recently explored to solve the OPF task with reduced computational burden; however, sufficient conditions for tightness of these relaxations are only available for restricted classes o…
Laboratory for Information and Decision Systems (LIDS)（信息与决策系统实验室（LIDS））；Massachusetts Institute of Technology（麻省理工学院）
We propose the first general and practical framework to design certifiable algorithms for robust geometric perception in the presence of a large amount of outliers. We investigate the use of a truncated least squares (TLS) cost function, which is known to be robust to outliers, but leads to hard, nonconvex, and nonsmooth optimization problems. Our first contribution is to show that -for a broad class of geometric perception problems- TLS estimation can be reformulated as an optimization over the ring of polynomials and Lasserre's hierarchy of convex moment relaxations is empirically tight at t…
We present a technique for producing valid dual bounds for nonconvex quadratic optimization problems. The approach leverages an elegant piecewise linear approximation for univariate quadratic functions due to Yarotsky, formulating this (simple) approximation using mixed-integer programming (MIP). Notably, the number of constraints, binary variables, and auxiliary continuous variables used in this formulation grows logarithmically in the approximation error. Combining this with a diagonal perturbation technique to convert a nonseparable quadratic function into a separable one, we present a mixe…
Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that immediately implies that the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the convex hull result holds whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. O…
An optimization problem considering AC power flow constraints and integer decision variables can usually be posed as a mixed-integer quadratically constrained quadratic program (MIQCQP) problem. In this paper, first, a set of valid linear equalities are applied to strengthen the semidefinite program (SDP) relaxation of the MIQCQP problem without significantly increasing the problem dimension so that an enhanced mixed-integer SDP (MISDP) relaxation, which is a mixed-integer convex problem, is obtained. Then, the enhanced MISDP relaxation is reformulated as a disjunctive programming (DP) problem…
This work considers the problem of locating a single source from noisy range measurements to a set of nodes in a wireless sensor network. We propose two new techniques that we designate as Source Localization with Nuclear Norm (SLNN) and Source Localization with l1-norm (SL-l1), which extend to arbitrary real dimensions, including 3D, our prior work on 2D source localization formulated in the complex plane. Broadly, our approach is based on formulating a Maximum-Likelihood (ML) estimation problem for the source position, and then using convex relaxation techniques to obtain a semidefinite prog…
We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $x\in\{0\}\cup[\ell,u]$, where $z$ is a binary indicator of $x\in[\ell,u]$ ($0 \leq \ell <u$), and $y$ "captures" $f(x)$, which is assumed to be convex and positive on its domain $[\ell,u]$, but otherwise $y=0$ when $x=0$. This model is very useful in nonlinear combinatorial optimization, where there is a fixed cost of operating an activity at level $x$ in the operating range $[\ell,u]$, and then there is a further (convex) variable cost $f(x)$. In particular, we study relaxations related to the perspective t…
Variable Aggregation-based Perspective Reformulation for Mixed-Integer Convex Optimization with Symmetry
School of Economics and Management, North China Electric Power University（华北电力大学经济与管理学院）；Thayer School of Engineering, Dartmouth College（达特茅斯学院塔克工程学院）；School of Economics and Management, University of Chinese Academy of Sciences（中国科学院大学经济与管理学院）；Department of Industrial and Systems Engineering, North Carolina State University（北卡罗来纳州立大学工业与系统工程系）
Verifying that input-output relationships of a neural network conform to prescribed operational specifications is a key enabler towards deploying these networks in safety-critical applications. Semidefinite programming (SDP)-based approaches to Rectified Linear Unit (ReLU) network verification transcribe this problem into an optimization problem, where the accuracy of any such formulation reflects the level of fidelity in how the neural network computation is represented, as well as the relaxations of intractable constraints. While the literature contains much progress on improving the tightne…
Joint object matching, also known as multi-image matching, namely, the problem of finding consistent partial maps among all pairs of objects within a collection, is a crucial task in many areas of computer vision. This problem subsumes bipartite graph matching and graph partitioning as special cases and is NP-hard, in general. We develop scalable linear programming (LP) relaxations with theoretical performance guarantees for joint object matching. We start by proposing a new characterization of consistent partial maps; this in turn enables us to formulate joint object matching as an integer li…
Given the noisy pairwise measurements among a set of unknown group elements, how to recover them efficiently and robustly? This problem, known as group synchronization, has drawn tremendous attention in the scientific community. In this work, we focus on orthogonal group synchronization that has found many applications, including computer vision, robotics, and cryo-electron microscopy. One commonly used approach is the least squares estimation that requires solving a highly nonconvex optimization program. The past few years have witnessed considerable advances in tackling this challenging prob…
A Tight SDP Relaxation for the Cubic-Quartic Regularization Problem
Xiangtan University（湘潭大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；University of California San Diego（加州大学圣地亚哥分校）；Hong Kong Baptist University（香港浸会大学）
