Semidefinite Programming Approach Research Articles

An economic-environmental energy management system design for MT-HVDC networks via a semi-definite programming approximation with robust analysis

This research deals with the economic-environmental dispatch problem (EEDP) in multi-terminal high-voltage direct-current (MT-HVDC) systems by proposing a convex approximation based on semi-definite programming (SDP). The exact formulation of the EEDP corresponds to a non-convex, nonlinear programming problem due to the presence of a nonlinear quadratic constraint regarding the products between voltage variables. The thermal plants' economic and objective functions are modeled using typical quadratic functions. The SDP approach allows reaching a convex approximation that ensures the global optimal solution for each objective function independently or the construction of the optimal Pareto front via the weighting-factor optimization methodology. The proposed SDP approach also considers uncertainties in the demand and in the available power of renewable sources, which makes it robust. The main contribution of this research is a multi-period analysis that includes large-scale renewable generation sources and a robust analysis regarding demand and variations in renewable generation. Numerical results in two MT-HVDC systems demonstrate the effectiveness of the proposed SDP approach when compared to combinatorial optimization algorithms. All numerical simulations were carried out using the CVX convex disciplined tool, with the help of the SEDUMI and SDPT solvers, in the MATLAB programming environment.

Scalable semidefinite programming approach to variational embedding for quantum many-body problems

In quantum embedding theories, a quantum many-body system is divided into localized clusters of sites which are treated with an accurate ‘high-level’ theory and glued together self-consistently by a less accurate ‘low-level’ theory at the global scale. The recently introduced variational embedding approach for quantum many-body problems combines the insights of semidefinite relaxation and quantum embedding theory to provide a lower bound on the ground-state energy that improves as the cluster size is increased. The variational embedding method is formulated as a semidefinite program (SDP), which can suffer from poor computational scaling when treated with black-box solvers. We exploit the interpretation of this SDP as an embedding method to develop an algorithm which alternates parallelizable local updates of the high-level quantities with updates that enforce the low-level global constraints. Moreover, we show how translation invariance in lattice systems can be exploited to reduce the complexity of projecting a key matrix to the positive semidefinite cone.

Variational learning algorithms for quantum query complexity

Quantum query complexity is pivotal in the analysis of quantum algorithms, encompassing well-known examples like search and period-finding algorithms. These algorithms typically involve a sequence of unitary operations and oracle calls dependent on an input variable. In this study, we introduce a variational learning approach to explore quantum query complexity. Our method employs an efficient parameterization of the unitary operations and utilizes a loss function derived from the algorithm’s error probability. We apply this technique to various quantum query complexities, notably devising a new algorithm that resolves the 5-bit Hamming modulo problem with four queries, addressing an open question from Cornelissen et al (2021 arXiv:2112.14682). This finding is corroborated by a semidefinite programming (SDP) approach. Our numerical method exhibits superior memory efficiency compared to SDP and can identify quantum query algorithms (QQAs) that require a smaller workspace register dimension, an aspect not optimized by SDP. These advancements present a significant step forward in the practical application and understanding of QQAs.

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.

On Solving MAX-SAT Using Sum of Squares

We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability (MAX-SAT) problem and weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiability problems by being competitive with some of the best solvers in the yearly MAX-SAT competition. Our solver combines sum of squares (SOS)–based SDP bounds and an efficient parser within a branch-and-bound scheme. On the theoretical side, we propose a family of semidefinite feasibility problems and show that a member of this family provides the rank-two guarantee. We also provide a parametric family of semidefinite relaxations for MAX-SAT and derive several properties of monomial bases used in the SOS approach. We connect two well-known SDP approaches for (MAX)-SAT in an elegant way. Moreover, we relate our SOS-SDP relaxations for partial MAX-SAT to the known SAT relaxations. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0036 .

Certifying the quantum Fisher information from a given set of mean values: a semidefinite programming approach

We introduce a semidefinite programming algorithm to find the minimal quantum Fisher information compatible with an arbitrary dataset of mean values. This certification task allows one to quantify the resource content of a quantum system for metrology applications without complete knowledge of the quantum state. We implement the algorithm to study quantum spin ensembles. We first focus on Dicke states, where our findings challenge and complement previous results in the literature. We then investigate states generated during the one-axis twisting dynamics, where in particular we find that the metrological power of the so-called multi-headed cat states can be certified using simple collective spin observables, such as fourth-order moments for small systems, and parity measurements for arbitrary system sizes.

Square Root Convexity of Fisher Information along Heat Flow in Dimension Two

Recently, Ledoux, Nair, and Wang proved that the Fisher information along the heat flow is log-convex in dimension one, that is d2dt2log(I(Xt))≥0 for n=1, where Xt is a random variable with density function satisfying the heat equation. In this paper, we consider the high dimensional case and prove that the Fisher information is square root convex in dimension two, that is d2dt2IX≥0 for n=2. The proof is based on the semidefinite programming approach.

Robust Hybrid Beamforming for Outage-Constrained Multigroup Multicast mmWave Transmission With Phase Shifter Impairments

This article investigates robust hybrid beamforming design for outage-constrained multigroup multicast millimeter-wave (mmWave) transmission. The phase shifter impairments incorporated in the analog beamformer can impose negative damage on the Quality of Service (QoS) of users. To address this imperfect hardware problem, phase shifter impairments are modeled, and robust hybrid beamforming is formulated to maximize the worst-case signal-to-interference–plus-noise ratio (SINR). The digital beamformer and analog beamformer are jointly optimized based on an alternating optimization (AO) approach, where both the outage probability of users and the average transmit power at the base station (BS) are derived into closed-form expressions in each iteration. Moreover, the semidefinite programming (SDP) approach is resorted to handling the hybrid beamforming design. Based on the penalty and the successive convex approximation (SCA) approaches, we further propose a penalty-SCA-SDP-AO (PSSA) algorithm to address the constant-modulus constraint caused by phase shifters and the rank-one constraint introduced by the semidefinite form. Simulation results show that compared to the conventional nonrobust design, our robust hybrid beamforming design generally improves 40% nonoutage performance at least.

Spherical Atomic Norm-Inspired Approach for Direction-of-Arrival Estimation of EM Waves Impinging on Spherical Antenna Array with Undefined Mutual Coupling

A spherical antenna array (SAA) is an array-designed arrangement capable of scanning in almost all the radiation sphere with constant directivity. It finds recent applications in aerospace, spacecraft, vehicular and satellite communications. Therefore, estimation of the direction-of-arrival (DoA) of electromagnetic (EM) waves that impinge on an SAA with unknown mutual coupling called for research attention. This paper proposed a spherical harmonic atomic norm minimization (SHANM) for DoA estimation using an SAA configuration. The gridless sparse signal recovery problem is considered in the spherical harmonic (SH) domain in conjunction with the atomic norm minimization (ANM). Because of the unavailability of the Vandermonde structure in the SH domain, the theorem of Vandermonde decomposition that is the mathematical basis of the traditional ANM methods finds no application in SH. Addressing this challenge, a low-dimensional semidefinite programming (SDP) approach implementing the SHANM method is developed. This approach is independent of Vandermonde decomposition, and directly recovers the atomic decomposition in SH. The numerical experimental results show the superior performance of the proposed method against the previous methods. In addition, accounting for the impacts of mutual coupling, an experimental measured data, which is the generally accepted ground of testing any method, is employed to illustrate the efficacy and robustness of the proposed methods. Finally, for achieving DoA estimation with sufficient localization accuracy using a SAA, the proposed SHANM-based method is a better option.

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.

Revisiting Semidefinite Programming Approaches to Options Pricing: Complexity and Computational Perspectives

In this paper, we consider the problem of finding bounds on the prices of options depending on multiple assets without assuming any underlying model on the price dynamics but only the absence of arbitrage opportunities. We formulate this as a generalized moment problem and utilize the well-known moment-sum-of-squares hierarchy of Lasserre to obtain bounds on the range of the possible prices. A complementary approach (also from Lasserre) is employed for comparison. We present several numerical examples to demonstrate the viability of our approach. The framework we consider makes it possible to incorporate different kinds of observable data, such as moment information, as well as observable prices of options on the assets of interest. History: Accepted by Antonio Frangioni, area editor for Design &amp; Analysis of Algorithms–Continuous. Funding: This work was supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement [Grant 813211 (POEMA)]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplementary Information [ https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1220 ] or is available from the IJOC GitHub software repository ( https://github.com/INFORMSJoC ) at [ http://dx.doi.org/10.5281/zenodo.6602361 ].

A semidefinite programming approach for the projection onto the cone of negative semidefinite symmetric tensors with applications to solid mechanics

We propose an algorithm for computing the projection of a symmetric second-order tensor onto the cone of negative semidefinite symmetric tensors with respect to the inner product defined by an assigned positive definite symmetric fourth-order tensor {mathcal {C}}. The projection problem is written as a semidefinite programming problem and an algorithm based on a primal-dual path-following interior point method coupled with a Mehrotra’s predictor-corrector approach is proposed. Implementations based on well-known symmetrization schemes and on direct methods are theoretically and numerically investigated taking into account tensors {mathcal {C}} arising in the modelling of masonry-like materials. For these special cases, indications on the preferable symmetrization scheme that take into account the conditioning of the arising linear systems are given.

Observer-based consensus for fractional-order multi-agent systems with positive constraint

This paper is concerned with consensus for fractional-order multi-agent systems by observer-type output feedback with positive constraint depicted by general linear model. First, an observer-based protocol is presented and the sufficient conditions for the consensus for fractional-order multi-agent systems with positive constraint are derived. Then, the simplified conditions without utilizing the detailed topology of the graph are given. In order to simplify the process of the solution of the feedback gain matrix and the observer gain matrix, the linear programming approach is used. When the system matrix is strictly Metzler, semidefinite programming approach is adopted. Finally, some illustration examples are presented as the verification of the obtained methods.

The background method: theory and computations.

The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader 'auxiliary function' framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh-Bénard convection between stress-free isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also outlined. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Joint pilot design and beamforming optimization in massive MIMO surveillance systems

This paper proposes a novel joint channel estimation and beamforming scheme for the massive multiple-input-multiple-output (MIMO) frequency-division duplexing (FDD) wireless legitimate surveillance system. With the proposed scheme, the monitor with the full duplex capability realizes the proactive eavesdropping of the suspicious link by leveraging the pilot attack approach. Specifically, exploiting the effective eavesdropping rate and the mean square error as performance metrics and setting a total power budget at the training and transmission phases, while guaranteeing the information from suspicious source can be successfully decode, joint pilot design, power allocation and beamforming strategy are formulated as optimization problems for the two objective functions: MSE and effective eavesdropping rate. A closed-form expression of the optimal pilot with the limited length can be obtained via the channel correlation. The optimal power problem at the training phase can be solved by a simple bisection method. Then, based on the obtained imperfect estimated channel, the jamming beamforming at monitor optimization algorithm is proposed by utilizing the convex Semidefinite Programming approach to maximize the effective eavesdropping rate. Numerical results show that the proposed joint pilot design, power allocation and beamforming optimization scheme can improve the surveillance performance of the legitimate monitor as compared to the existing passive eavesdropping and jamming-assisted eavesdropping.

An SDP-based approach for computing the stability number of a graph

Finding the stability number of a graph, i.e., the maximum number of vertices of which no two are adjacent, is a well known NP-hard combinatorial optimization problem. Since this problem has several applications in real life, there is need to find efficient algorithms to solve this problem. Recently, Gaar and Rendl enhanced semidefinite programming approaches to tighten the upper bound given by the Lovász theta function. This is done by carefully selecting some so-called exact subgraph constraints (ESC) and adding them to the semidefinite program of computing the Lovász theta function. First, we provide two new relaxations that allow to compute the bounds faster without substantial loss of the quality of the bounds. One of these two relaxations is based on including violated facets of the polytope representing the ESCs, the other one adds separating hyperplanes for that polytope. Furthermore, we implement a branch and bound (B&B) algorithm using these tightened relaxations in our bounding routine. We compare the efficiency of our B&B algorithm using the different upper bounds. It turns out that already the bounds of Gaar and Rendl drastically reduce the number of nodes to be explored in the B&B tree as compared to the Lovász theta bound. However, this comes with a high computational cost. Our new relaxations improve the run time of the overall B&B algorithm, while keeping the number of nodes in the B&B tree small.

Exploiting perturbed and coalescent anchor node geometry with semidefinite relaxation for sensor network localization

Localization in wireless sensor networks (WSNs) suffer from performance issues whenever the anchor nodes (which are aware of their location) are subjected to motion from their usual position. Moreover, accurate localization demands more anchor nodes which is a scarce resource and needs to be used judiciously. In the current work, we propose a novel framework that addresses these two prime concerns by harnessing the inter relationship of anchor node geometry. For an unknown source node surrounded by anchor nodes, the anchors lying on the inner boundary of the deployment geometry may be carrying closely related information about source node, leading to redundancy and inefficient utilization. By anticipating the level of correlation between these anchors, localization can be made more frugal. Rigorous mathematical analysis is carried out to derive lower bounds on estimated locations. Based on fisher information from two proposed models, a convex estimation objective function is formulated using semidefinite programming (SDP) approach to validate the theoretical proceedings. Based on the findings, the proposed method is able to successfully extract useful information about the unknown source node location with limited number of anchor nodes, hence achieving superior localization.

Positivity in Multifield Effective Field Theories.

We discuss the general method for obtaining full positivity bounds on multifield effective field theories (EFTs). While the leading order forward positivity bounds are commonly derived from the elastic scattering of two (superposed) external states, we show that, for a generic EFT containing three or more low-energy modes, this approach only gives incomplete bounds. We then identify the allowed parameter space as the dual to a spectrahedron, constructed from crossing symmetries of the amplitude, and show that finding the optimal bounds for a given number of modes is equivalent to a geometric problem: finding the extremal rays of a spectrahedron. We show how this is done analytically for simple cases and numerically formulated as semidefinite programming (SDP) problems for more complicated cases. We demonstrate this approach with a number of well-motivated examples in particle physics and cosmology, including EFTs of scalars, vectors, fermions, and gravitons. In all these cases, we find that the SDP approach leads to results that either improve the previous ones or are completely new. We also find that the SDP approach is numerically much more efficient.

Improved RSSD-Based Source Localization With Unknown Sensor Position Errors

In this letter, a received signal strength difference (RSSD) approach is presented to localize a source with unknown transmit power in the presence of sensor position errors. The performance of conventional least squares (LS) algorithms is degraded because they consider measurement errors rather than estimation errors. An algorithm is presented here to overcome this problem. First, a robust minimax mean squared error (MSE) estimator is developed based on the estimation error for bounded location estimation and sensor position errors to minimize the worst case sum of the variance and squared norm of the bias. This nonlinear problem is solved by transforming the nonconvex objective function into a convex optimization problem using the S-procedure, relaxation, and semidefinite programming. This problem is extended to the unknown path loss exponent case. Necessary and sufficient conditions are given for convergence of the proposed RSSD convex relaxation of MSE semidefinite programming (RCRM-SDP) approach. Simulation results are presented which confirm the robustness of this method for sufficiently large signal to noise ratios (SNRs).

Success-or-Draw: A Strategy Allowing Repeat-Until-Success in Quantum Computation.

The repeat-until-success strategy is a standard method to obtain success with a probability that grows exponentially with the number of iterations. However, since quantum systems are disturbed after a quantum measurement, how to perform repeat-until-success strategies in certain quantum algorithms is not straightforward. In this Letter, we propose a new structure for probabilistic higher-order transformation named success-or-draw, which allows a repeat-until-success implementation. For that we provide a universal construction of success-or-draw structure that works for any probabilistic higher-order transformation on unitary operations. We then present a semidefinite programming approach to obtain optimal success-or-draw protocols and analyze in detail the problem of inverting a general unitary operation.