Equivalent Semidefinite Programming Research Articles

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

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

Efficient 2D DOA Estimation via Decoupled Projected Atomic Norm Minimization

This paper presents an efficient two-dimensional (2D) direction of arrival (DOA) estimation method, termed as decoupled projected atomic norm minimization (D-PANM), to solve the angle-ambiguity problem. It first introduces a novel atomic metric via projecting the original atom set onto a smoothing space, based on which we formulate an equivalent semi-definite programming (SDP) problem. Then, two relatively low-complexity decoupled Toeplitz matrices can be obtained to estimate the DOAs. We further exploit the structural information hidden in the newly constructed data to avoid pair matching for the azimuth and elevation angles when the number of sensors is odd, and then propose a fast and feasible decoupled alternating projections (D-AP) algorithm, reducing computational complexity to a great extent. Numerical simulations are performed to demonstrate that the proposed algorithm is no longer restricted by angle ambiguity scenarios, but instead provides a more stable estimation performance, even when multiple signals share the same angles in both azimuth and elevation dimensions. Additionally, it greatly improves the resolution, with control of the computation load compared with the existing atomic norm minimization (ANM) algorithm.

Piecewise SOS-convex moment optimization and applications via exact semi-definite programs

This paper presents exact Semi-Definite Program (SDP) reformulations for infinite-dimensional moment optimization problems involving a new class of piecewise Sum-of-Squares (SOS)-convex functions and projected spectrahedral support sets. These reformulations show that solving a single SDP finds the optimal value and an optimal probability measure of the original moment problem. This is done by establishing an SOS representation for the non-negativity of a piecewise SOS-convex function over a projected spectrahedron. Finally, as an application and a proof-of-concept illustration, the paper presents numerical results for the Newsvendor and revenue maximization problems with higher-order moments by solving their equivalent SDP reformulations. These reformulations promise a flexible and efficient approach to solving these models. The main novelty of the present work in relation to the recent research lies in finding the solution to moment problems, for the first time, with piecewise SOS-convex functions from their numerically tractable exact SDP reformulations.

Robust international portfolio optimization with worst‐case mean‐CVaR

We propose a robust international portfolio optimization model in a worst-case mean-CVaR framework. In our model, we assume that the distributions, the first- and second-order moments of returns of assets and exchange rates are ambiguous. To control the conservatism of our robust model, we incorporate a new support set constructed by intervals of deviations from the no-arbitrage condition in currency markets into the ambiguity set of distribution. Our model can be reformulated as an equivalent semi-definite programming problem, which is computationally tractable. We conduct empirical experiments by the weekly rolling window strategy during the total period, the 1997 Asian Financial Crisis period, and the recent stable period. Using various performance measures, we investigate the out-of-sample performance of our model with comparison to those of other four benchmark models. The experimental results demonstrate that our model has the best performance in terms of return and various risk adjusted return measures during all of the three periods. They suggest that investors can obtain significant benefits when employing the robust portfolio strategy and the new support set, and considering the ambiguity of the first- and second-order moments during both volatile and stable periods.

Robust International Portfolio Optimization with Worst-Case Mean-LPM

This paper proposes a robust international portfolio optimization model with the consideration of worst-case lower partial moment (LPM) and worst-case mean return. In our model, we assume that the distributions and the first- and second-order moments of distributions of returns of assets and exchange rates are all ambiguous. The proposed model can be reformulated into an equivalent semidefinite programming (SDP) problem, which is computationally tractable. For investigation of the performance of our model, we also give two benchmark models. The first benchmark model is a scenario-based model which uses historical observations of returns to approximate the future distributions. The second benchmark model only considers the ambiguity of distributions but does not consider the ambiguity of the first- and second-order moments of distributions. We conduct empirical experiments in a rolling forward way to evaluate the out-of-sample performances of our proposed model, the two benchmark models, and an equally weighted model using the return measures and various risk-adjusted return measures. The result shows that our model has the best performance. It verifies that investors can obtain benefits when employing the robust model and considering the ambiguity of the first- and second-order moments of distributions.

Target-based distributionally robust optimization for single machine scheduling

In the traditional single-machine scheduling problem with uncertain processing time, distributionally robust optimization models are established in situations where the distribution information is not known exactly. One well-studied robust optimization method used in scheduling is the distributionally β-robust optimization technique. The goal of this technique is to maximize the probability that the total flow time of the job sequence does not exceed the predetermined target level T. However, it is possible that the excess value of the total flow time beyond the target T may be incredibly large, and therefore it is necessary to consider the optimized excess value. To address this concern, two new target-based risk measures are introduced. First, target-oriented CVaR, the worst-case conditional value-at-risk (CVaR) is used to minimize the excess value beyond T, which accounts for both the excess probability and the excess value. Second, the essential riskiness index (ERI) is extended to a multi-piecewise linear function for excess value control in single-machine scheduling problems. To obtain a tractable closed-form solutions of the risk measures, the CVaR-based measure is decomposed into an assignment problem and a second-order cone programming problem, and the ERI is converted to its equivalent semi-definite programming and second-order cone counterparts. To efficiently solve the risk measures for single-machine scheduling, a bisection search algorithm and a Benders decomposition algorithm are designed. In the numerical analysis, the excess value and the probability are investigated with respect to two distributions, namely the two-point distribution and the truncated normal distribution. The experiment design also considers different ranges of expectation and variance, tolerance levels, and problem scales. It is discovered that the CVaR and ERI-based methods are always better than the distributionally β-robust scheduling approach in terms of excess value.

Two Relaxation Methods for Rank Minimization Problems

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be lifted to give an equivalent semidefinite program with complementarity constraints. The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We develop two relaxations and show that constraint qualification holds at any stationary point of either relaxation of the rank minimization problem, and we explore the structure of the local minimizers.

A Distributionally Robust Optimization Model for Real-Time Power Dispatch in Distribution Networks

The integration of large-scale renewable distributed generation (DG) units in distribution networks brings new challenges for real-time power dispatch, including active power and reactive power (P-Q) coupling and operational uncertainties. To reduce the conservativeness of robust optimization (RO) solutions, this paper proposes a distributionally robust realtime power dispatch model, which characterizes the uncertainty of DG output with the information of first-order and second-order moments. Furthermore, a two-stage scheme, which includes economic dispatch (ED) and corrective control, is incorporated in the model, in which ED optimizes the active power schedule and corrective control strategies of P-Q are generated to eliminate overvoltage problems. The proposed model can be transformed into an equivalent semidefinite programming problem and solved via the delayed constraint generation algorithm. Numerical test results show that the proposed model reduces operational costs and mitigates overvoltage problems significantly compared with the conventional one-stage model. The proposed model also outperforms the conventional RO model by exploiting probability information to achieve statistically optimal dispatch.

T-optimal designs for multi-factor polynomial regression models via a semidefinite relaxation method

We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models where the design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets. Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When the regression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtain an equivalent semidefinite program. When there are two or more factors in the models, we apply a moment relaxation technique and approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When the relaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and its optimality can be additionally confirmed by an equivalence theorem. The methodology is illustrated with several examples.

Robust Transceiver Design in Full-Duplex MIMO Cognitive Radios

We consider a full duplex (FD) multiple-input multiple-output (MIMO) underlay cognitive radio cellular network, in which an FD secondary base-station (BS) serves multiple half-duplex uplink (UL) and downlink secondary users (SUs) at the same time and frequency. We assume that the channel state information (CSI) available at the transmitters is imperfect, and the errors of the CSI are assumed to be norm bounded. Under the impact of channel uncertainty, we address the sum mean-squared-errors minimization problem subject to individual power constraints at the UL SUs, a total power-constraint at the secondary BS, and the interference constraints on the primary users by the secondary network. By transforming the problem into an equivalent semidefinite programming (SDP), we propose an iterative alternating algorithm to compute the transceiver matrices jointly. Moreover, to reduce the high computational complexity of the SDP method, we develop a cutting-set method, which solves the problem by alternating between an optimization step (transceiver design) and a pessimization step (worst-case channel analysis). Numerical results are presented to show the effectiveness and robustness of the proposed algorithms.

Partial facial reduction: simplified, equivalent SDPs via approximations of the PSD cone

We develop a practical semidefinite programming (SDP) facial reduction procedure that utilizes computationally efficient approximations of the positive semidefinite cone. The proposed method simplifies SDPs with no strictly feasible solution (a frequent output of parsers) by solving a sequence of easier optimization problems and could be a useful pre-processing technique for SDP solvers. We demonstrate effectiveness of the method on SDPs arising in practice, and describe our publicly-available software implementation. We also show how to find maximum rank matrices in our PSD cone approximations (which helps us find maximal simplifications), and we give a post-processing procedure for dual solution recovery that generally applies to facial-reduction-based pre-processing techniques. Finally, we show how approximations can be chosen to preserve problem sparsity.

On the Lovász theta function and some variants

The Lovász theta function of a graph is a well-known upper bound on the stability number. It can be computed efficiently by solving a semidefinite program (SDP). Actually, one can solve either of two SDPs, one due to Lovász and the other to Grötschel et al. The former SDP is often thought to be preferable computationally, since it has fewer variables and constraints. We derive some new results on these two equivalent SDPs. The surprising result is that, if we weaken the SDPs by aggregating constraints, or strengthen them by adding cutting planes, the equivalence breaks down. In particular, the Grötschel et al. scheme typically yields a stronger bound than the Lovász one.

Semi-Infinite Programming using High-Degree Polynomial Interpolants and Semidefinite Programming

In a common formulation of semi-infinite programs, the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval. If this function is sufficiently closely approximable by a polynomial or a rational function, then the semi-infinite program can be reformulated as an equivalent semidefinite program, which in turn can be solved with interior point methods very efficiently to high accuracy. On the other hand, solving this semidefinite program is challenging if the polynomials involved are of high degree, due to numerical difficulties and bad scaling arising both from the polynomial approximations and from the fact that the semidefinite programming constraints coming from the sum-of-squares representation of nonnegative polynomials are badly scaled. We combine polynomial function approximation techniques and polynomial programming to overcome these numerical difficulties, using sum-of-squares interpolants. Specifically, it is shown that the c...

Distributionally robust hydro-thermal-wind economic dispatch

With the penetration of wind energy increasing, uncertainty has become a major challenge in power system dispatch. Hydro power can change rapidly and is regarded as one promising complementary energy resource to mitigate wind power fluctuation. Joint scheduling of hydro, thermal, and wind energy is attracting more and more attention nowadays. This paper proposes a distributionally robust hydro-thermal-wind economic dispatch (DR-HTW-ED) method to enhance the flexibility and reliability of power system operation. In contrast to the traditional stochastic optimization (SO) and adjustable robust optimization (ARO) method, distributionally robust optimization (DRO) method describes the uncertain wind power output by all possible probability distribution functions (PDFs) with the same mean and variance recovered from the forecast data, and optimizes the expected operation cost in the worst distribution. Traditional DRO optimized the random parameter in entire space, which is sometimes contradict to the actual situation. In this paper, we restrict the wind power uncertainty in a bounded set, and derive an equivalent semi-definite programming (SDP) for the DR-HTW-ED using S-lemma. A delayed constraint generation algorithm is suggested to solve it in a tractable manner. The proposed DR-HTW-ED is compared with the existing ARO based hydro-thermal-wind economic dispatch (AR-HTW-ED). Their respective features are shown from the perspective of computational efficiency and conservativeness of dispatch strategies.

Robust chance-constrained support vector machines with second-order moment information

Support vector machines (SVM) is one of the well known supervised classes of learning algorithms. Basic SVM models are dealing with the situation where the exact values of the data points are known. This paper studies SVM when the data points are uncertain. With some properties known for the distributions, chance-constrained SVM is used to ensure the small probability of misclassification for the uncertain data. As infinite number of distributions could have the known properties, the robust chance-constrained SVM requires efficient transformations of the chance constraints to make the problem solvable. In this paper, robust chance-constrained SVM with second-order moment information is studied and we obtain equivalent semidefinite programming and second order cone programming reformulations. The geometric interpretation is presented and numerical experiments are conducted. Three types of estimation errors for mean and covariance information are studied in this paper and the corresponding formulations and techniques to handle these types of errors are presented.

On the Interference Tolerance of the Primary System in Cognitive Radio Networks

In this letter, we consider underlay spectrum access in a cognitive radio network, where two coexisting cellular systems, i.e., primary and cognitive, share the same frequency band. The throughput performance of this system is directly linked to the tolerated interference level at the primary receiver. We investigate the fundamental trade-off between the interference tolerance in the primary system and the achievable throughput of the cognitive network. We formulate the network performance as a multi-objective optimization problem with a set of objectives including minimizing the interference imposed on each primary user, and maximizing the intended signal received at every secondary user. We then derive an equivalent standard semi-definite programming form and prove that the derived problem always yields rank-one solutions. The obtained solution set provides the best achievable throughput (characterized by the maximum intended signal power at each cognitive user) for a given level of interference tolerance in the primary system. It also gives the achievable gain on the system throughput if the primary system is able to compromise and increase its interference tolerance, such framework could be a base for negotiations between the operators. An improvement of 0.38 bits/s/channel-use is observed on the achievable throughput for 1 dB higher interference tolerance at the primary receivers.

A semi-definite relaxation approach for partial discharge source location in transformers

Current issues on localization algorithms based on the time difference of arrival (TDOA) for the partial discharge (PD) source include its sensitivity to time delay error, easy local convergence or divergence, and the large amount of computational load and time. A semi-definite relaxation method for PD source location to solve the time delay positioning equations is proposed in this paper, which using the semi-definite programming problem has the characteristic that can ensure obtained the global optimal solution. The proposed method converts nonlinear time delay equations into an equivalent semi-definite programming (SDP) problem by equivalent transformation and rank-1 relaxation firstly. Then use the interior point algorithm to solve the SDP problem that to obtain a unique global optimum solution, and extract a rank-1 component from the global optimal solution of relaxed SDP. Finally, to serve as a good approximate of the original problem for the PD location. The method was used to localize a measured PD source signal in the laboratory, and the results were compared with those of positioning by the Newton-iterative method. The comparison showed that the method can reduce sensitivity for the time delay error as well as effectively solve TDOA location equations, thereby ensuring that the result is a unique global optimal solution with high positioning efficiency. The localization algorithm problem is solved with the unavoidable and difficult-to-locate time delay error.

Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization

The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust LSP or SOCP under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time.

Models and algorithms for distributionally robust least squares problems

We present three different robust frameworks using probabilistic ambiguity descriptions of the data in least squares problems. These probability ambiguity descriptions are given by: (1) confidence region over the first two moments; (2) bounds on the probability measure with moments constraints; (3) the Kantorovich probability distance from a given measure. For the first case, we give an equivalent formulation and show that the optimization problem can be solved using a semidefinite optimization reformulation or polynomial time algorithms. For the second case, we derive the equivalent Lagrangian problem and show that it is a convex stochastic programming problem. We further analyze three special subcases: (i) finite support; (ii) measure bounds by a reference probability measure; (iii) measure bounds by two reference probability measures with known density functions. We show that case (i) has an equivalent semidefinite programming reformulation and the sample average approximations of case (ii) and (iii) have equivalent semidefinite programming reformulations. For ambiguity description (3), we show that the finite support case can be solved by using an equivalent second order cone programming reformulation.

Robust Quadratic Regression and Its Application to Energy-Growth Consumption Problem

We propose a robust quadratic regression model to handle the statistics inaccuracy. Unlike the traditional robust statistic approaches that mainly focus on eliminating the effect of outliers, the proposed model employs the recently developed robust optimization methodology and tries to minimize the worst-case residual errors. First, we give a solvable equivalent semidefinite programming for the robust least square model with ball uncertainty set. Then the result is generalized to robust models underl1- andl∞-norm critera with general ellipsoid uncertainty sets. In addition, we establish a robust regression model for per capital GDP and energy consumption in the energy-growth problem under the conservation hypothesis. Finally, numerical experiments are carried out to verify the effectiveness of the proposed models and demonstrate the effect of the uncertainty perturbation on the robust models.