Semidefinite Programming Formulation Research Articles

Construction of ellipsoid convex model of bounded uncertainties with outlier detection for application in non-probabilistic topology optimization

The ellipsoid convex model can be suitably used for structural analysis and design optimization under uncertain-but-bounded parameters and loads. Such a model can be constructed using measured samples of the uncertainties. However, the presence of outliers is often unavoidable due to system fluctuations in the measurements. Thus, it is necessary to detect any outliers among the samples before constructing the uncertainty model to prevent over-conservativeness. To this end, the present paper proposes a rational approach for constructing ellipsoid convex models with outlier detection. The concept of the local outlier factor (LOF) is utilized, in conjunction with the k-nearest natural neighbor to adaptively determine the neighborhood range. Then the outliers are detected using the scaled median absolute deviation method, based on the LOF values obtained. Finally, the minimum-volume ellipsoid convex model is constructed with a mathematically strict and efficient semi-definite programming formulation using the normal samples. The validity of the proposed approach is demonstrated with numerical examples. The application of the constructed uncertainty model in non-probabilistic robust topology optimization is demonstrated, and the results show the effectiveness of the proposed approach.

Semi-definite programming and quantum information

Abstract This paper presents a comprehensive exploration of semi-definite programming (SDP) techniques within the context of quantum information. It examines the mathematical foundations of convex optimization, duality, and SDP formulations, providing a solid theoretical framework for addressing optimization challenges in quantum systems. By leveraging these tools, researchers and practitioners can characterize classical and quantum correlations, optimize quantum states, and design efficient quantum algorithms and protocols. The paper also discusses implementational aspects, such as solvers for SDP and modeling tools, enabling the effective employment of optimization techniques in quantum information processing. The insights and methodologies presented in this paper have proven instrumental in advancing the field of quantum information, facilitating the development of novel communication protocols, self-testing methods, and a deeper understanding of quantum entanglement.

Maximizing Social Welfare and Agreement via Information Design in Linear-Quadratic-Gaussian Games

Information design in an incomplete information game involves a designer that aims to influence players' actions through signals generated from a designed probability distribution to optimize its objective function. For quadratic design objective functions, if the players have quadratic payoffs that depend on the players' actions and an unknown payoff-relevant state, and signals on the state that follow a Gaussian distribution conditional on the state realization, the information design problem is a semidefinite program (SDP) [1]. In this note, we seek to characterize the optimal information design analytically by leveraging the SDP formulation, when the design objective is to maximize social welfare or the agreement among players' action. We show that full information disclosure maximizes social welfare when there is a common payoff state, the payoff dependencies among players' actions are homogeneous, or when the signals are public. When the objective is to maximize the agreement among players' actions, not revealing any information is optimal. When the objective is a weighted combination of social welfare and agreement terms, we establish a threshold weight below which full information disclosure is optimal under public signals for games with homogeneous payoffs. Numerical results corroborate the analytical results, and identify partial information disclosure structures that are optimal.

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.

Source Localization Using RSS Measurements with Sensor Position Uncertainty

Received signal strength- (RSS-) based localization has attracted considerable attention for its low cost and easy implementation. In plenty of existing work, sensor positions, which play an important role in source localization, are usually assumed perfectly known. Unfortunately, they are often subject to uncertainties, which directly leads to effect on localization result. To tackle this problem, we study the RSS-based source localization with sensor position uncertainty. Sensor position uncertainty will be modeled as two types: Gaussian random variable and unknown nonrandom variable. For either of the models, two semidefinite programming (SDP) methods are proposed, i.e., SDP-1 and SDP-2. The SDP-1 method proceeds from the nonconvex problem with respect to the maximum likelihood (ML) estimation and then obtains an SDP problem using proper approximation and relaxation. The SDP-2 method first transfers the sensor position uncertainties to the source position and then obtains an SDP formulation following a similar idea as in SDP-1 method. Numerical examples demonstrate the performance superiority of the proposed methods, compared to some existing methods assuming perfect sensor position information.

Multi-output multilevel best linear unbiased estimators via semidefinite programming

Multifidelity forward uncertainty quantification (UQ) problems often involve multiple quantities of interest and heterogeneous models (e.g., different grids, equations, dimensions, physics, surrogate and reduced-order models). While computational efficiency is key in this context, multi-output strategies in multilevel/multifidelity methods are either sub-optimal or non-existent. In this paper we extend multilevel best linear unbiased estimators (MLBLUE) to multi-output forward UQ problems and we present new semidefinite programming formulations for their optimal setup. Not only do these formulations yield the optimal number of samples required, but also the optimal selection of low-fidelity models to use. While existing MLBLUE approaches are single-output only and require a non-trivial nonlinear optimization procedure, the new multi-output formulations can be solved reliably and efficiently. We demonstrate the efficacy of the new methods and formulations in practical UQ problems with model heterogeneity.

Gridless wideband DOA estimation in nonuniform noise with increased DOFs

This paper presents gridless schemes for direction finding of wideband sources using sparse array with nonuniform noise. According to the frequencies relationship, we obtain multiple equivalent scaled difference coarrays, which are combined later to formulate synthetic virtual array when sources share the same normalized power spectrum in different subbands. After treating the term affected by nonuniform noise as hole, an atomic norm optimization problem to recover the noise-free signal of interpolated synthetic uniform linear array (ULA) is designed. We analyze the estimation error distribution of the noise-independent terms in synthetic virtual array and based on this, impose a constraint to avoid tuning process and effect of noise. Furthermore, it extremely increases the number of detectable sources, even making it possible to exceed the degree of freedoms of difference coarray. For the general case when sources have arbitrary powers, a multi-snapshot atomic norm optimization problem is formulated, where signal from scaled coarrays is considered as partial observation of synthetic ULA for different snapshots. Similarly, a newly constructed constraint is added, with which most of the above advantages are sustained. Besides, we give the semidefinite programming form and the Cramer-Rao bounds. The superiority of the proposed algorithms is demonstrated by simulations.

Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming

The systematic design of exact optimal designs of experiments is typically challenging, as it results in nonconvex optimization problems. The literature on the computation of model-based exact optimal designs of experiments via mathematical programming, when the covariates are categorical variables, is still scarce. We propose mixed-integer semidefinite programming formulations, to find exact D-, A- and I-optimal designs for linear models, and locally optimal designs for nonlinear models when the design domain is a finite set of points. The strategy requires: (i) the generation of a set of candidate treatments; (ii) the formulation of the optimal design problem as a mixed-integer semidefinite program; and (iii) its solution, employing appropriate solvers. For comparison, we use semidefinite programming-based formulations to find equivalent approximate optimal designs. We demonstrate the application of the algorithm with various models, considering both unconstrained and constrained setups. Equivalent approximate optimal designs are used for comparison.

Distributionally Robust Risk Map for Learning-Based Motion Planning and Control: A Semidefinite Programming Approach

In this article, we propose a novel safety specification tool, called the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributionally robust risk map</i> (DR-risk map), for a mobile robot operating in a learning-enabled environment. Given the robot's position, the map aims to reliably assess the conditional value-at-risk (CVaR) of collision with obstacles whose movements are inferred by Gaussian process regression (GPR). Unfortunately, the inferred distribution is subject to errors, making it difficult to accurately evaluate the CVaR of collision. To overcome this challenge, our tool measures the risk under the worst-case distribution in a so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ambiguity set</i> that characterizes allowable distribution errors. To resolve the infinite-dimensionality issue inherent in the construction of the DR-risk map, we derive a tractable semidefinite programming formulation that provides an upper bound of the risk, exploiting techniques from modern distributionally robust optimization. As a concrete application for motion planning, a distributionally robust RRT* algorithm is considered using the risk map that addresses distribution errors caused by GPR. Furthermore, a motion control method is devised using the DR-risk map in a learning-based model predictive control (MPC) formulation. In particular, a neural network approximation of the risk map is proposed to reduce the computational cost in solving the MPC problem. The performance and utility of the proposed risk map are demonstrated through simulation studies that show its ability to ensure the safety of mobile robots despite learning errors.

Policy Synthesis for Switched Linear Systems With Markov Decision Process Switching

In this article, we study the synthesis of mode switching protocols for a class of discrete-time switched linear systems in which the mode jumps are governed by Markov decision processes (MDPs). We call such systems MDP-JLS for brevity. Each state of the MDP corresponds to a mode in the switched system. The probabilistic state transitions in the MDP represent the mode transitions. We focus on finding a policy that selects the switching actions at each mode such that the switched system is guaranteed to be stable. Given a policy in the MDP, the considered MDP-JLS reduces to a Markov jump linear system (MJLS). We consider both mean-square stability and stability with probability one. For mean-square stability, we leverage existing stability conditions for MJLSs and propose efficient semidefinite programming formulations to find a stabilizing policy in the MDP. For stability with probability one, we derive new sufficient conditions and compute a stabilizing policy using linear programming. We also extend the policy synthesis results to MDP-JLS with uncertain mode transition probabilities.

A gridless method for direction finding with sparse arrays in nonuniform noise

The performance of direction finding methods would deteriorate due to unknown nonuniform noise. To cope with this problem, we propose a novel gridless direction finding method based on atomic norm minimization exploiting sparse linear array in nonuniform noise. Specifically, after eliminating the concentrated nonuniform noise related term in coarray signal, the concept of array interpolation is used to recover both the noiseless counterpart of the removed term as well as the holes in coarray. Thus, the effect of nonuniform noise is removed. Besides, we impose a new constraint based on the estimation error distribution of the noise independent terms in the coarray signal. The regularization parameter can thus be selected directly from the Chi-square distribution probability table. In the proposed method, the tedious selection of regularization parameter and the effect of grid mismatch are avoided. Moreover, we derive the corresponding semidefinite programming (SDP) form. With its optimal solution, eigen-decomposition with high complexity is avoided for subsequent DOA estimation. Different from the traditional SDP form, it has an additional transformation matrix composed of the estimation error. Simulations show that the proposed method owns the highest estimation accuracy than previous algorithms in the nonuniform noise.

Real-Time Flexibility Quantification of a Building HVAC System for Peak Demand Reduction

The quantification of heating, ventilation, and air condition (HVAC) system flexibility is critical to the operations of both the grid and buildings in demand response (DR) programs. However, the flexibility quantification is challenging due to the non-linearity and non-convexity of thermal dynamics associated with HVAC components. This paper proposes a novel HVAC flexibility quantification method based on a semidefinite programming (SDP) formulation. The SDP is reformulated from the non-convex problem of HVAC power optimization, and can be solved efficiently in real-time. The physics-based HVAC model is incorporated to ensure the reliability and accuracy of solutions. The quantification results are organized into an HVAC flexibility table that can provide response strategies on adjusting HVAC setpoints in response to the grid signals received. The developed response strategies minimize occupant discomfort while satisfying grid requirements. A case study of a test building model is carried out to illustrate the flexibility quantification framework and compares the performance of two DR strategies.

Identifying 3D Genome Organization in Diploid Organisms via Euclidean Distance Geometry

The spatial organization of the DNA in the cell nucleus plays an important role for gene regulation, DNA replication, and genomic integrity. Through the development of chromosome conformation capture experiments (such as 3C, 4C, Hi-C) it is now possible to obtain the contact frequencies of the DNA at the whole-genome level. In this paper, we study the problem of reconstructing the 3D organization of the genome from such whole-genome contact frequencies. A standard approach is to transform the contact frequencies into noisy distance measurements and then apply semidefinite programming (SDP) formulations to obtain the 3D configuration. However, neglected in such reconstructions is the fact that most eukaryotes including humans are diploid and therefore contain two copies of each genomic locus. We prove that the 3D organization of the DNA is not identifiable from distance measurements derived from contact frequencies in diploid organisms. In fact, there are infinitely many solutions even in the noise-free setting. We then discuss various additional biologically relevant and experimentally measurable constraints (including distances between neighboring genomic loci and higher-order interactions) and prove identifiability under these conditions. Furthermore, we provide SDP formulations for computing the 3D embedding of the DNA with these additional constraints and show that we can recover the true 3D embedding with high accuracy from both noiseless and noisy measurements. Finally, we apply our algorithm to real pairwise and higher-order contact frequency data and show that we can recover known genome organization patterns.

Distribution Grid Optimal Power Flow (D-OPF): Modeling, Analysis, and Benchmarking

In the power distribution systems, optimal power flow (D-OPF) is formulated as a non-convex and non-linear programming (NLP) problem. Convex relaxation and linear approximation models have been increasingly adopted to achieve computational efficiency for D-OPF. Despite the benefits of scalability and global optimality, each method is based on certain assumptions, performs differently, and may lead to solutions that are physically not meaningful. In this context, this work numerically evaluates the relative performance of second-order cone programming (SOCP), semi-definite programming (SDP), and linear programming (LP) formulations of D-OPF in terms of their feasibility, optimality, and scalability with respect to NLP-based formulations. We also compare the bus injection (in bus voltage and current variables) and branch flow (in active and reactive power flow variables) based on NLP formulations. The performance is evaluated using small (123-node), medium (730-node), and large (2522-node) sized distribution feeders. Case studies, which are backed up by visualization of the analytical models for the solution space to the extent possible, show that (1) the feasibility and exactness of relaxed D-OPF formulations depend upon the problem type, (2) some NLP formulations are computationally more tractable than others, (3) different NLP formulations can converge to different local solutions, and (4) the approximate linear model may underestimate or overestimate the cost function (depending upon the problem type) and may lead to AC-infeasible solutions.

Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy

The design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is usually a polynomial function of design variables, allowing us to build an equivalent non-linear semidefinite programming formulation over a semi-algebraic feasible set. This formulation is subsequently solved using the Lasserre moment-sum-of-squares hierarchy, generating a sequence of outer convex approximations that monotonically converges from below to the optimum of the original problem. Globally optimal solutions can subsequently be extracted using the Curto-Fialkow flat extension theorem. Furthermore, we show that a simple correction to the solutions of the relaxed problems establishes a feasible upper bound, thereby deriving a simple sufficient condition of global $\varepsilon$-optimality. When the original problem possesses a unique minimum, we show that this solution is found with a zero optimality gap in the limit. These theoretical findings are illustrated on several examples of topology optimization of frames and shells, for which we observe that the hierarchy converges in a finite (rather small) number of steps.

Resilience in Multirobot Multitarget Tracking With Unknown Number of Targets Through Reconfiguration

We address the problem of maintaining resource availability in a networked multirobot team performing distributed tracking of an unknown number of targets in a bounded environment. Robots are equipped with sensing and computational resources, enabling them to cooperatively track a set of targets using a distributed probability hypothesis density (PHD) filter. We use the trace of a robot's sensor measurement noise covariance matrix to quantify its sensing quality. While executing the tracking task, if a robot experiences sensor quality degradation, the team's communication network is reconfigured such that the robot with the faulty sensor may share information with other robots to improve the team's target-tracking ability without enforcing a large change in the number of active communication links. A central system monitor executes the network reconfiguration computations. We consider two different PHD fusion methods and propose four different mixed-integer semi-definite programming (MISDP) formulations (two formulations for each PHD fusion method) to accomplish our objective. All MISDP formulations are validated in simulation.

Transient security control for bulk power system with selection of aggregated distributed energy resources

Amongst the background of the demand of modern society for renewables and decarbonisation, the gradual penetration of distributed energy resources (DER) has been observed. While many DER studies have centered on reliability and economic issues, there is a scarcity of research on the possible grid support that aggregated DER could provide to bulk electric systems (BES). In the present paper, a novel control scheme acting with optimally selected DER aggregations (DERAs) is proposed to prevent transmission cascading from overcurrent protection in transient process. The security control scheme is obtained by reshaping a transient energy function, where the energy function not only characterises the nominal dynamic behaviour of the power system transient, but also depicts transmission tripping caused by the overcurrent. On the basis of the availability and capability thereof, the participating DERAs are selected via a sparse matrix, which is solved with linear matrix inequalities (LMIs) in the form of semi-definite programming (SDP) without the linearisation of the system model. Test examples are provided with IEEE standard 14 and 118 Bus Systems.

Semi-device-independent certification of entanglement in superdense coding

Superdense coding is a paradigmatic protocol in quantum information science, employing a quantum communication channel to send classical information more efficiently. As we show here, it can be understood as a particular case of a prepare and measure experiment, a scenario that has attracted growing attention for its fundamental and practical applications. Formulating superdense coding as a prepare and measure scenario allows us to provide a semi-device-independent witness of entanglement that significantly improves over previous tests. Furthermore, we also show how to adapt our results into self-testing of maximally entangled states and also provide a semidefinite program formulation allowing to efficiently optimize, for any shared quantum state, the probability of success in the superdense coding protocol.

Addressing the Conditional and Correlated Wind Power Forecast Errors in Unit Commitment by Distributionally Robust Optimization

In this paper, a study of the day-ahead unit commitment problem with stochastic wind power generation is presented, which considers conditional, and correlated wind power forecast errors through a distributionally robust optimization approach. Firstly, to capture the characteristics of random wind power forecast errors, the least absolute shrinkage, and selection operator (Lasso) is utilized to develop a robust conditional error estimator, while an unbiased estimator is used to obtain the covariance matrix. The conditional error, and the covariance matrix are then used to construct an enhanced ambiguity set. Secondly, we develop an equivalent mixed integer semidefinite programming (MISDP) formulation of the two-stage distributionally robust unit commitment model with a polyhedral support of random variables. Further, to efficiently solve this problem, a novel cutting plane algorithm that makes use of the extremal distributions identified from the second-stage semidefinite programming (SDP) problems is introduced. Finally, numerical case studies show the advantage of the proposed model in capturing the spatiotemporal correlation in wind power generation, as well as the economic efficiency, and robustness of dispatch decisions.

Nonconvex lifted unbalanced branch flow model: Derivation, implementation and experiments

This paper develops a novel nonconvex formulation of the unbalanced power flow equations. This formulation extends the lifted nonconvex ‘DistFlow’ a.k.a. balanced branch flow model formulation to the unbalanced case. The feasible set is characterized by linear and nonconvex quadratic equations in matrix variables. It is shown that this formulation is equivalent to previously published rank-constrained semidefinite programming formulations. The formulation is implemented and compared numerically with rectangular and polar versions of the nonlinear AC unbalanced optimal power flow problem.