Semidefinite Programming Research Articles

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.

Differentially Private Sketch-and-Solve for Community Detection via Semidefinite Programming

Differentially Private Sketch-and-Solve for Community Detection via Semidefinite Programming

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.

A Second-Order Cone Programming (SOCP) Based Optimal Power Flow (OPF) Model With Cyclic Constraints for Power Transmission Systems

For meshed power networks, even though the conic relaxation is shown to be exact, the relaxation of angles may not be exact using the existing Second-Order Cone Programming (SOCP) based optimal power flow (OPF) models. Power transmission networks generally have meshed orientation, and the cyclic angle constraints are not satisfied with the existing SOCP-OPF models. This work proposes a SOCP-OPF model for power transmission networks that satisfies the cyclic angle constraints for any mesh in the network. The novelty of the proposed OPF model is that it defines a convex envelope to represent the relative bus voltage angles that satisfy the cyclic constraint criteria for a meshed network. The proposed SOCP-OPF model is tested on the IEEE 14-bus, 57-bus, 118-bus, 500-bus, and 2736-bus networks. The case studies demonstrate that the proposed model is computationally efficient and scalable for large transmission networks compared to the Nonlinear Programming (NLP) and semi-definite programming (SDP) counterparts.

Variational determination of the two‐electron reduced density matrix: A tutorial review

AbstractThe two‐electron reduced density matrix (2RDM) carries enough information to evaluate the electronic energy of a many‐electron system. The variational 2RDM (v2RDM) approach seeks to determine the 2RDM directly, without knowledge of the wave function, by minimizing this energy with respect to variations in the elements of the 2RDM, while also enforcing known N‐representability conditions. In this tutorial review, we provide an overview of the theoretical underpinnings of the v2RDM approach and the N‐representability constraints that are typically applied to the 2RDM. We also discuss the semidefinite programming (SDP) techniques used in v2RDM computations and provide enough Python code to develop a working v2RDM code that interfaces to the libSDP library of SDP solvers.This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Software > Quantum Chemistry

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.

Hydrazine Concentration Estimation in the Secondary Circuit of a Nuclear Power Plant: a IQC LPV approach

This paper addresses the design of a new linear parameter varying estimator, to estimate the hydrazine molar concentration in the secondary circuit of a nuclear power plant. The method uses dynamic shaping filters of arbitrary order, to formulate the estimation performance objectives. The theory is based on L2-gain synthesis, based on uncertainty characterization via static integral quadratic constraint multipliers. It is shown that the problem can be formulated as a semi definite programming problem, that can be solved efficiently using numerical solvers. The proposed approach is applied to real data, coming from a nuclear power plant located in France. The obtained results demonstrates the efficiency of the proposed approach.

Robust next-day scheduling of PV generation sources supplying a standalone DC microgrid via a semi-definite programming model

Robust next-day scheduling of PV generation sources supplying a standalone DC microgrid via a semi-definite programming model

Frequency-Domain Identification of Discrete-Time Systems using Sum-of-Rational Optimization

This paper proposes a new computationally tractable method to fit coefficients of a fixed-order discrete-time transfer function to a measured frequency response, with stability guaranteed. The problem is formulated as a non-convex global sum-of-rational optimization problem whose objective function is the sum of weighted squared residuals at each observed frequency datapoint. Stability is enforced using a polynomial matrix inequality constraint. The problem is solved by a moment-sum-of-squares hierarchy of semidefinite programs through a framework for sum-of-rational-functions optimization. Convergence of the moment-sum-of-squares program is guaranteed as the bound on the degree of the sum-of-squares polynomials approaches infinity. The performance of the proposed method is demonstrated using numerical simulation examples.

Reduced-Order Neural Network Synthesis With Robustness Guarantees.

In the wake of the explosive growth in smartphones and cyber-physical systems, there has been an accelerating shift in how data are generated away from centralized data toward on-device-generated data. In response, machine learning algorithms are being adapted to run locally on board, potentially hardware-limited, devices to improve user privacy, reduce latency, and be more energy efficient. However, our understanding of how these device-orientated algorithms behave and should be trained is still fairly limited. To address this issue, a method to automatically synthesize reduced-order neural networks (having fewer neurons) approximating the input-output mapping of a larger one is introduced. The reduced-order neural network's weights and biases are generated from a convex semidefinite program that minimizes the worst case approximation error with respect to the larger network. Worst case bounds for this approximation error are obtained and the approach can be applied to a wide variety of neural networks architectures. What differentiates the proposed approach to existing methods for generating small neural networks, e.g., pruning, is the inclusion of the worst case approximation error directly within the training cost function, which should add robustness to out-of-sample data points. Numerical examples highlight the potential of the proposed approach. The overriding goal of this article is to generalize recent results in the robustness analysis of neural networks to a robust synthesis problem for their weights and biases.

Semi-Definite Programming Based Scalable and Accurate Optimal Power Flow Models for Radial Distribution Networks

Semi-Definite Programming Based Scalable and Accurate Optimal Power Flow Models for Radial Distribution Networks

Reach-Avoid Verification Based on Convex Optimization

In this paper we propose novel sufficient conditions for verifying reach-avoid properties of continuous-time systems modelled by ordinary differential equations (ODEs). Given a system, an initial set, a safe set and a target set of states, we say that the reach-avoid property holds, if for all initial conditions in the initial set, any trajectory of the system starting at them will eventually, i.e. in unbounded yet finite time, enter the target set while remaining inside the safe set until that first target hit (that is, if the system starting from the initial set can reach the target set safely). Based on a discount value function, two sets of quantified constraints are derived for verifying the reach-avoid property via the computation of exponential/asymptotic guidance-barrier functions (they form a barrier escorting the system to the target set safely at an exponential or asymptotic rate). It is interesting to find that one set of constraints whose solution is termed exponential guidance-barrier functions is just a simplified version of the existing one derived from the moment based method, while the other one whose solution is termed asymptotic guidance-barrier functions is completely new. Furthermore, built upon this new set of constraints, we derive a set of more expressive constraints, which includes the aforementioned two sets of constraints as special instances, providing more chances for verifying the reach-avoid property successfully. Finally, several examples demonstrate the theoretical developments and performance of proposed sufficient conditions using semi-definite programming methods.

Semidefinite programming approximation for a matrix optimization problem over an uncertain linear system

Semidefinite programming approximation for a matrix optimization problem over an uncertain linear system

Semi-Supervised Clustering of Sparse Graphs: Crossing the Information-Theoretic Threshold.

The stochastic block model is a canonical random graph model for clustering and community detection on network-structured data. Decades of extensive study on the problem have established many profound results, among which the phase transition at the Kesten-Stigum threshold is particularly interesting both from a mathematical and an applied standpoint. It states that no estimator based on the network topology can perform substantially better than chance on sparse graphs if the model parameter is below a certain threshold. Nevertheless, if we slightly extend the horizon to the ubiquitous semi-supervised setting, such a fundamental limitation will disappear completely. We prove that with an arbitrary fraction of the labels revealed, the detection problem is feasible throughout the parameter domain. Moreover, we introduce two efficient algorithms, one combinatorial and one based on optimization, to integrate label information with graph structures. Our work brings a new perspective to the stochastic model of networks and semidefinite program research.

Towards targeted exploration for non-stochastic disturbances

We present a novel targeted exploration strategy for linear time-invariant systems without stochastic assumptions on the noise, i.e., without requiring independence or zero mean, allowing for deterministic model misspecifications. This work utilizes classical data-dependent uncertainty bounds on the least-squares parameter estimates in the presence of energy-bounded noise. We provide a sufficient condition on the exploration data that ensures a desired error bound on the estimated parameter. Using common approximations, we derive a semidefinite program to compute the optimal sinusoidal input excitation. Finally, we highlight the differences and commonalities between the developed non-stochastic targeted exploration strategy and conventional exploration strategies based on classical identification bounds through a numerical example.

The exact worst-case convergence rate of the alternating direction method of multipliers

AbstractRecently, semidefinite programming performance estimation has been employed as a strong tool for the worst-case performance analysis of first order methods. In this paper, we derive new non-ergodic convergence rates for the alternating direction method of multipliers (ADMM) by using performance estimation. We give some examples which show the exactness of the given bounds. We also study the linear and R-linear convergence of ADMM in terms of dual objective. We establish that ADMM enjoys a global linear convergence rate if and only if the dual objective satisfies the Polyak–Łojasiewicz (PŁ) inequality in the presence of strong convexity. In addition, we give an explicit formula for the linear convergence rate factor. Moreover, we study the R-linear convergence of ADMM under two scenarios.

Convex Combination for Wireless Localization Using Biased RSS Measurements

Received signal strength- (RSS-) based localization in wireless sensor networks (WSNs) has attracted significant attention due to its advantages of low cost and simple implementation. In practice, RSS measurements may suffer from sensor biases, which deteriorates the localization accuracy. However, most of the existing localization methods are designed for bias-free measurements. In this paper, we propose a convex combination method for RSS localization in the presence of sensor biases. The parameter vector composed of unknown location and sensor biases is estimated simultaneously by using a convex combination of some virtual points. These virtual points form a convex hull, into which the parameter vector falls with large probability. By this, the original nonconvex estimation problem is converted to be convex. Numerical examples demonstrate the superiority of the proposed method in terms of localization accuracy, compared to the existing semidefinite programming (SDP) methods.

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.

Quantifying channel coherence via the norm distance

Quantifying the number of resources contained in a physical object has been one of the core topics in the resource theory of coherence. In this paper, we introduce the dynamical coherence measures based on a class of norms in the classical channel setting. It is proved that it satisfies faithfulness, decreases monotonically under the maximally incoherent superchannels, and is convex. Moreover, we show that it satisfies subadditivity under both the composition and tensor product of channels. Especially, the diamond measure as a special case is discussed in detail, it can reduce to trace norm of coherence, satisfies amortization inequality, and can be calculated efficiently using a semidefinite program. In addition, we introduce the creation-coherent diamond measure and find that neither the detection coherence nor the creation coherence of a channel exceeds the coherence of the channel, which does not exceed the purity of the channel. Second, we introduce the corresponding dephasing measure, which is a dynamical coherence measure under the dephasing-covariant incoherent superchannels. Meanwhile, we also introduce the dephasing diamond measure as a special case. Third, we use the dephasing diamond measure to accurately calculate the coherence values of some important noisy channels such as amplitude damping channel, phase damping channel, and depolarizing channel, respectively, and give the sufficient and necessary conditions for an unital qubit channel with a parameter probability vector to be a coherent channel. Finally, the operational interpretation of our diamond measure in the binary channel discrimination task is investigated.

High uncertainty aware localization and error optimization of mobile nodes for wireless sensor networks

<p>The localization of mobile sensor nodes in a wireless sensor network (WSN) is a key research area for the speedy development of wireless communication and microelectronics. The localization of mobile sensor nodes massively depends upon the received signal strength (RSS). Recently, the least squared relative error (LSRE) measurements are optimized using traditional semidefinite programming (SDP) and the location of the mobile sensor nodes was determined using the previous localization methods like least squared relative error and semidefinite programming (LSRE-SDP), and approximate nonlinear least squares and semidefinite programming (ANLS-SDP). Therefore, in this work, a novel high uncertainty aware-localization error correction and optimization (HUA-LECO) model is employed to minimize the aforementioned problems regarding the localization of mobile sensor nodes and enhance the performance efficiency of root mean square error (RMSE) results. Here, the position of target mobile sensor nodes is evaluated based on the gathered measurements while discarding faulty data. Here, an iterative weight updation approach is utilized to perform localization based on Monte Carlo simulations. Simulation results show significant improvement in terms of RMSE results in comparison with traditional LSRE-SDP and ANLS-SDP methods under high uncertainty.</p>