Semidefinite Programming Relaxation Research Articles

Generalized Semi-infinite Polynomial Optimization and Semidefinite Programming Relaxations

Generalized Semi-infinite Polynomial Optimization and Semidefinite Programming Relaxations

Stochastic dual dynamic programming for optimal power flow problems under uncertainty

Planning in the power sector has to take into account the physical laws of alternating current (AC) power flows as well as uncertainty in the data of the problems, both of which greatly complicate optimal decision making. We propose a computationally tractable framework to solve multi-stage stochastic optimal power flow (OPF) problems in AC power systems. Our approach uses recent results on dual convex semi-definite programming (SDP) relaxations of OPF problems in order to adapt the stochastic dual dynamic programming (SDDP) algorithm for problems with a Markovian structure. We show that the usual SDDP lower bound remains valid and that the algorithm converges to a globally optimal policy of the stochastic AC-OPF problem as long as the SDP relaxations are tight. To test the practical viability of our approach, we set up a case study of a storage siting, sizing, and operations problem. We show that the convex SDP relaxation of the stochastic problem is usually tight and discuss ways to obtain near-optimal physically feasible solutions when this is not the case. The algorithm finds a physically feasible policy with an optimality gap of 3% and yields a significant added value of 27% over a rolling deterministic policy, which leads to overly optimistic policies and underinvestment in flexibility. This suggests that the common industry practice of assuming direct current and deterministic problems should be reevaluated by considering models that incorporate realistic AC flows and stochastic elements in the data as potentially more realistic alternatives.

The circle packing problem: A theoretical comparison of various convexification techniques

We consider the problem of packing a given number of congruent circles with the maximum radius in a unit square as a mathematical optimization problem. Due to the presence of non-overlapping constraints, this problem is a notoriously difficult nonconvex quadratically constrained optimization problem, which possesses many local optima. We consider several popular convexification techniques, giving rise to linear programming relaxations and semidefinite programming relaxations for the circle packing problem. We compare the strength of these relaxations theoretically, thereby proving the conjectures by Anstreicher. Our results serve as a theoretical justification for the ineffectiveness of existing machinery for convexification of non-overlapping constraints.

Certifiably optimal rotation and pose estimation based on the Cayley map

We present novel, convex relaxations for rotation and pose estimation problems that can a posteriori guarantee global optimality for practical measurement noise levels. Some such relaxations exist in the literature for specific problem setups that assume the matrix von Mises-Fisher distribution (a.k.a., matrix Langevin distribution or chordal distance) for isotropic rotational uncertainty. However, another common way to represent uncertainty for rotations and poses is to define anisotropic noise in the associated Lie algebra. Starting from a noise model based on the Cayley map, we define our estimation problems, convert them to Quadratically Constrained Quadratic Programs (QCQPs), then relax them to Semidefinite Programs (SDPs), which can be solved using standard interior-point optimization methods; global optimality follows from Lagrangian strong duality. We first show how to carry out basic rotation and pose averaging. We then turn to the more complex problem of trajectory estimation, which involves many pose variables with both individual and inter-pose measurements (or motion priors). Our contribution is to formulate SDP relaxations for all these problems based on the Cayley map (including the identification of redundant constraints) and to show them working in practical settings. We hope our results can add to the catalogue of useful estimation problems whose solutions can be a posteriori guaranteed to be globally optimal.

Solving Two-stage Quadratic Multiobjective Problems via Optimality and Relaxations

This paper focuses on the study of robust two-stage quadratic multiobjective optimization problems. We formulate new necessary and sufficient optimality conditions for a robust two-stage multiobjective optimization problem. The obtained optimality conditions are presented by means of linear matrix inequalities and thus they can be numerically validated by using a semidefinite programming problem. The proposed optimality conditions can be elaborated further as second-order conic expressions for robust two-stage quadratic multiobjective optimization problems with separable functions and ellipsoidal uncertainty sets. We also propose relaxation schemes for finding a (weak) efficient solution of the robust two-stage multiobjective problem by employing associated semidefinite programming or second-order cone programming relaxations. Moreover, numerical examples are given to demonstrate the solution variety of our flexible models and the numerical verifiability of the proposed schemes.

Active Reconfigurable Intelligent Surface Enhanced Internet of Medical Things.

The incredible potentiality of reconfigurable intelligent surface (RIS) in addressing power supply and obstacle environment of Internet of Medical Things (IoMT) has been capturing our interest. Considering the nettlesome "double-fading" effect introduced by passive RIS, we investigate an active RIS-enhanced IoMT system in this article, where the wireless power transfer (WPT) from power station (PS) to IoMT devices and the wireless information transfer (WIT) from IoMT devices to the access point (AP) are both implemented with the assistance of active RIS. Aiming to maximize the sum throughput of the considered IoMT system, a joint design of time schedules and reflecting coefficient matrices of the active RIS is proposed. Trapped by the non-convex and obstinate optimization problem, we explore the semi-definite programming (SDP) relaxation and successive convex approximation (SCA) techniques based on alternating optimization (AO) algorithm. Simulation results verify our solution approach to the intractable optimization problem and showcase the boosted spectrum and energy efficiency of the active RIS-enhanced IoMT system.

On the tightness of an SDP relaxation for homogeneous QCQP with three real or four complex homogeneous constraints

On the tightness of an SDP relaxation for homogeneous QCQP with three real or four complex homogeneous constraints

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.

A Semidefinite Programming Method Weighted Sum Based for Solving a Multi-area Emission (Environmental) Economic Dispatch Problem

This paper presents a semi-definite programming (SDP) optimisation approach for solving multi area, emission/economic dispatch (MAEED) problems in thermal power systems. The multi-objective problem was transformed into matrix form as an SDP relaxation problem and subsequently solved with a MATLAB programming suite. The system inequality and equality constraints were entered into a Self-Dual Minimisation (SeDuMi) parser. Simulations were tested on four (4) IEEE benchmark models to validate the efficiency of the proposed SDP method. However, in the generation of the Pareto-fr ont solutions, ideal minimum points were used in the determination of the maximum spread out of the Pareto solutions by the algorithm. This involves the use of a standard weighted sum method in generating the Pareto-optimal solution between two objective functions. Twenty-one (21) runs were carried out for the parameter value to explore the effect of control weight selection k1. A comparative study demonstrated the effectiveness of tie-line constraint and without tie-line constraint, respectively on the optimi sation model for the power dispatch problems solved by using the proposed SDP approach. The optimisation results for the minimum fuel cost using the SDP approach for the case when a tie-line constraint is included was US$ 2,151.7 /hr with minimum emission of 3.4687 Ton/hr. This is lower than US$ 2,162.2 /hr with minimum emission of 3.6662 Ton/hr obtained when the tie-line constraint is not included in the optimisation model on IEEE four-area, sixteen network. Additionally, the comparative results with all other methods reported in the literature for different test cases were also presented in this paper. Lower operational costs were obtained for both tie-lines and no tie-lines power transfer cases, showing that the SDP method performed well in accuracy compared to other methods. The SDP method was employed in this research to deal with the multi-area emission economic power dispatch problem due to its inherent advantages in achieving a global optimal solution and, this approach would also assist the power administrator to come up with an improved decision making in attaining an optimum power dispatch of power generation.

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.