Semidefinite Programming Research Articles

Hierarchy of Quadratic Lyapunov Functions for Linear Time-Varying and Related Systems

A method for constructing homogeneous polynomial Lyapunov functions is presented for linear time-varying or switched-linear systems and the class of nonlinear systems that can be represented as such. The method uses a simple recursion based on the Kronecker product to generate a hierarchy of related dynamical systems, whose first element is the system under study and the second element is the well-known Lyapunov differential equation. It is then proven that a quadratic Lyapunov function for the system at one level in the hierarchy, which can be found via semidefinite programming, is a homogeneous polynomial Lyapunov function for the system at the base level in the hierarchy. Searching for Lyapunov functions of the foregoing kind is equivalent to searching for homogeneous polynomial Lyapunov functions via the formulation of sum-of-squares programs. The quadratic perspective presented in this paper enables the easy development of procedures to compute bounds on pointwise-in-time system metrics, such as peak norms, system stability margins, and many other performance measures. The applications of the theory to analyzing an aircraft model, on the one hand, and an experimental aerospace vehicle, on the other hand, are presented. The theory can be comprehended with a first course on state-space control systems and an elementary knowledge of convex programming.

Communication and Energy Efficient Decentralized Learning Over D2D Networks

Device-to-device (D2D)-assisted decentralized learning has been proposed for mobile devices to collaboratively train artificial intelligence networks without the centralized parameter server. However, a densely connected network will cause large learning latency and energy consumption due to the limited computation and communication resources. In addition, link selection and aggregation weight have a significant impact on the learning performance. To cope with these challenges, we propose a joint computing power adjustment, wireless resource allocation, link selection, and aggregation weight adaptation mechanism to improve both communication and energy efficiencies. Specifically, the learning performances including the convergence rate, per-iteration learning latency, and per-iteration energy consumption are first analyzed. Then, an optimization problem is formulated to minimize the total learning cost, which is defined as the weighted sum of total learning latency and energy consumption. Given a network topology, the computing power and wireless resource allocation are optimized by the alternating optimization algorithm. Moreover, the optimal aggregation weight is obtained by semidefinite programming. With respect to link selection, we propose a tabu search based meta-heuristic algorithm to approximately achieve feasible solutions with a low computational complexity. Finally, extensive experiments demonstrate that the proposed link selection algorithm can significantly reduce the learning cost under the given learning accuracy requirement.

Secret key rate bounds for quantum key distribution with faulty active phase randomization

Decoy-state quantum key distribution (QKD) is undoubtedly the most efficient solution to handle multi-photon signals emitted by laser sources, and provides the same secret key rate scaling as ideal single-photon sources. It requires, however, that the phase of each emitted pulse is uniformly random. This might be difficult to guarantee in practice, due to inevitable device imperfections and/or the use of an external phase modulator for phase randomization in an active setup, which limits the possible selected phases to a finite set. Here, we investigate the security of decoy-state QKD when the phase is actively randomized by faulty devices, and show that this technique is quite robust to deviations from the ideal uniformly random scenario. For this, we combine a novel parameter estimation technique based on semi-definite programming, with the use of basis mismatched events, to tightly estimate the parameters that determine the achievable secret key rate. In doing so, we demonstrate that our analysis can significantly outperform previous results that address more restricted scenarios.

Intelligent Reflecting Surface-Assisted Wireless Powered Heterogeneous Networks

In this paper, we introduce an intelligent reflecting surface (IRS)-assisted wireless powered heterogeneous network (WPHN) consisting of two heterogeneous groups of devices. Specifically, one group of devices, i.e., energy-harvesting devices (EHDs), are charged by external energy supplies, while the other group of devices, i.e., non-energy-harvesting devices (NEHDs), are powered by internal energy supplies. An IRS aims to participate in the wireless energy transfer (WET) in downlink and the wireless information transfer (WIT) in the uplink. A sum throughput maximization problem is formulated subject to the constraints of individual energy consumption, transmission time scheduling, and IRS phase shifts. To cope with the non-convexity of the problem, we first derive the optimal IRS phase shifts of the uplink WIT independently. Next, the semi-definite programming (SDP) relaxation is adopted to recast this non-convex problem into the convex one, which can be numerically solved. Then, a novel low-complexity scheme is developed to gain more insights and mitigate the computational complexity induced by the SDP relaxation. In particular, the dual problem and Karush-Kuhn-Tucker conditions are first utilized to obtain the optimal transmission time scheduling. Then, we propose a method based on Riemannian manifold optimization to compute the optimal IRS phase shifts of the downlink WET in closed-form. Finally, simulation results are presented to verify the optimality of our proposed scheme, and highlight the benefits induced by the IRS to coordinate these heterogeneous devices.

An approximation algorithm for high-dimensional table compression on balanced [formula omitted]-partite graph

This paper considers the high-dimensional table compression problem on balanced k-partite graph. The objective is to choose half of the vertices from each of the k-partite to maximize the total weight of edges connecting the chosen vertices. Our main contribution is a 0.8785-approximation algorithm for the high-dimensional table compression problem by introducing the α-independent solutions through Lasserre semidefinite programming. This new algorithm improved two previous low-dimensional results, namely the 0.8731-approximation algorithm of Wu et al. for the one-dimensional case and the 0.6708-approximation of Xu and Du for the two-dimensional case.

A Decision Rule Approach for Two-Stage Data-Driven Distributionally Robust Optimization Problems with Random Recourse

We study two-stage stochastic optimization problems with random recourse, where the coefficients of the adaptive decisions involve uncertain parameters. To deal with the infinite-dimensional recourse decisions, we propose a scalable approximation scheme via piecewise linear and piecewise quadratic decision rules. We develop a data-driven distributionally robust framework with two layers of robustness to address distributional uncertainty. We also establish out-of-sample performance guarantees for the proposed scheme. Applying known ideas, the resulting optimization problem can be reformulated as an exact copositive program that admits semidefinite programming approximations. We design an iterative decomposition algorithm, which converges under some regularity conditions, to reduce the runtime needed to solve this program. Through numerical examples for various known operations management applications, we demonstrate that our method produces significantly better solutions than the traditional sample-average approximation scheme especially when the data are limited. For the problem instances for which only the recourse cost coefficients are random, our method exhibits slightly inferior out-of-sample performance but shorter runtimes compared with a competing approach. History: Accepted by Nicola Secomandi, Area Editor for Stochastic Models & Reinforcement Learning. Funding: This work was supported by the National Science Foundation [Grants 2342505 and 2343869]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0306 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0306 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Two-Dimensional Super-Resolution Direction-of-Arrival Estimation for Arbitrary Planar Array Geometries

The two-dimensional grid-free compressed sensing method with planar microphone arrays is expected to achieve super-resolution direction-of-arrival (DOA) estimation for acoustic sources in the hemispherical space in front of array. The existing methods are not yet well compatible with arbitrary planar array geometries. In this paper, we aim to break this limitation and explore a novel method applicable to arbitrary planar array geometries. First, the covariance between source-induced signals at arbitrary two microphones is formulated as related to the two-dimensional inverse Fourier transform of the source strength. Then, a mathematical model with a source sparsity constraint is built in the continuous domain to denoise the covariances of measured signals, which can be solved by decoupled positive semidefinite programming. Finally, the DOAs are extracted from the solution of the mathematical model via a matrix pencil and pairing method. Simulation and experimental results demonstrate the effectiveness of the proposed method.

On scaling robust feedback control and state estimation problems in power networks

Many mainstream robust control/estimation algorithms for power networks are designed using the Lyapunov theory as it provides performance guarantees for linear/nonlinear models of uncertain power networks but comes at the expense of scalability and sensitivity. In particular, Lyapunov-based approaches rely on forming semi-definite programs (SDPs) that are (i) not scalable and (ii) extremely sensitive to the choice of the bounding scalar that ensures the strict feasibility of the linear matrix inequalities (LMIs). This paper addresses these two issues by employing a celebrated non-Lyapunov approach (NLA) from the control theory literature. In lieu of linearized models of power grids, we focus on (the more representative) nonlinear differential algebraic equation (DAE) models and showcase the simplicity, scalability, and parameter-resiliency of NLA. For some power systems, the approach is nearly fifty times faster than solving SDPs via standard solvers with almost no impact on the performance. The case studies also demonstrate that NLA can be applied to more realistic scenarios in which (i) only partial state data is available and (ii) sparsity structures are imposed on the feedback gain. The paper also showcases that virtually no degradation in state estimation quality is experienced when applying NLA.

Approximating sparse quadratic programs

Given a matrix A∈Rn×n, we consider the problem of maximizing xTAx subject to the constraint x∈{−1,1}n. This problem, called MaxQP by Charikar and Wirth [FOCS'04], generalizes MaxCut and has natural applications in data clustering and in the study of disordered magnetic phases of matter. Charikar and Wirth showed that the problem admits an Ω(1/lg⁡n) approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC'05] showed that the same approach yields an Ω(1) approximation when A corresponds to a graph of bounded chromatic number. Both these results rely on solving the semidefinite relaxation of MaxQP, whose currently best running time is O˜(n1.5⋅min⁡{N,n1.5}), where N is the number of nonzero entries in A and O˜ ignores polylogarithmic factors.In this sequel, we abandon the semidefinite approach and design purely combinatorial approximation algorithms for special cases of MaxQP where A is sparse (i.e., has O(n) nonzero entries). Our algorithms are superior to the semidefinite approach in terms of running time, yet are still competitive in terms of their approximation guarantees. More specifically, we show that:•MaxQP admits a (1/2Δ)-approximation in O(nlg⁡n) time, where Δ=O(1) is the maximum degree of the corresponding graph.•Unit MaxQP, where A∈{−1,0,1}n×n, admits a (1/2d)-approximation in O(n) time when the corresponding graph is d-degenerate, and a (1/3δ)-approximation in O(n1.5) time when the corresponding graph has δn edges for δ=O(1).•MaxQP admits a (1−ε)-approximation in O(n) time when the corresponding graph and each of its minors have bounded local treewidth.•Unit MaxQP admits a (1−ε)-approximation in O(n2) time for H-minor free graphs.

Variational quantum state discriminator for supervised machine learning

Abstract Quantum state discrimination (QSD) is a fundamental task in quantum information processing with numerous applications. We present a variational quantum algorithm that performs the minimum-error QSD, called the variational quantum state discriminator (VQSD). The VQSD uses a parameterized quantum circuit that is trained by minimizing a cost function derived from the QSD, and finds the optimal positive-operator valued measure (POVM) for distinguishing target quantum states. The VQSD is capable of discriminating even unknown states, eliminating the need for expensive quantum state tomography. Our numerical simulations and comparisons with semidefinite programming demonstrate the effectiveness of the VQSD in finding optimal POVMs for minimum-error QSD of both pure and mixed states. In addition, the VQSD can be utilized as a supervised machine learning algorithm for multi-class classification. The area under the receiver operating characteristic curve obtained in numerical simulations with the Iris flower dataset ranges from 0.97 to 1 with an average of 0.985, demonstrating excellent performance of the VQSD classifier.

Tighter Bounds on Transient Moments of Stochastic Chemical Systems

The use of approximate solution techniques for the Chemical Master Equation is a common practice for the analysis of stochastic chemical systems. Despite their widespread use, however, many such techniques rely on unverifiable assumptions and only a few provide mechanisms to control the approximation error quantitatively. Addressing this gap, Dowdy and Barton (J Chem Phys 149(7):074103, 2018) proposed an optimization-based technique for the computation of guaranteed bounds on the moment trajectories associated with stochastic chemical systems, thereby providing a general framework for rigorous uncertainty quantification. Here, we present an extension of this method. The key contribution is a new hierarchy of convex necessary moment conditions that build upon partitioning of the time domain. These conditions reflect the temporal causality that is inherent to the moment trajectories associated with stochastic processes described by the Chemical Master Equation and can be strengthened by simple refinement of the time domain partition. Analogous to the original method, these conditions generate a hierarchy of semidefinite programs that furnishes monotonically improving bounds on the trajectories of the moments and related statistics of stochastic chemical systems. Compared to its predecessor, the presented hierarchy produces bounds that are at least as tight and features new bound tightening mechanisms such as refinement of the time domain partition which often enable the computation of dramatically tighter bounds with lower computational cost. We analyze the properties of the presented hierarchy, discuss some aspects of its practical implementation and demonstrate its merits with several examples.

Graph-based conditions for feedback stabilization of switched and LPV systems

This paper presents novel stabilizability conditions for switched linear systems with arbitrary and uncontrollable underlying switching signals. We distinguish and study two particular settings: (i) the robust case, in which the active mode is completely unknown and unobservable, and (ii) the mode-dependent case, in which the controller depends on the current active switching mode. The technical developments are based on graph-theory tools, relying in particular on the path-complete Lyapunov functions framework. The main idea is to use directed and labeled graphs to encode Lyapunov inequalities to design robust and mode-dependent piecewise linear state-feedback controllers. This results in novel and flexible conditions, with the particular feature of being in the form of linear matrix inequalities (LMIs). Our technique thus provides a first controller-design strategy allowing piecewise linear feedback maps and piecewise quadratic (control) Lyapunov functions by means of semidefinite programming. Numerical examples illustrate the application of the proposed techniques, the relations between the graph order, the robustness, and the performance of the closed loop.

New lower bounds on crossing numbers of Km,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{m,n}$$\\end{document} from semidefinite programming

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph Km,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{m,n}$$\\end{document}, extending a method from de Klerk et al. (SIAM J Discrete Math 20:189–202, 2006) and the subsequent reduction by De Klerk, Pasechnik and Schrijver (Math Prog Ser A and B 109:613–624, 2007). We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that cr(K10,n)≥4.87057n2-10n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathop {\ extrm{cr}}\\limits (K_{10,n}) \\ge 4.87057 n^2 - 10n$$\\end{document}, cr(K11,n)≥5.99939n2-12.5n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathop {\ extrm{cr}}\\limits (K_{11,n}) \\ge 5.99939 n^2-12.5n$$\\end{document}, cr(K12,n)≥7.25579n2-15n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathop {\ extrm{cr}}\\limits (K_{12,n}) \\ge 7.25579 n^2 - 15n$$\\end{document}, cr(K13,n)≥8.65675n2-18n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathop {\ extrm{cr}}\\limits (K_{13,n}) \\ge 8.65675 n^2-18n$$\\end{document} for all n. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.

Preventive–corrective stability constrained OPF in microgrids for accommodating DG plug-and-play

To accommodate unpredictable plug-and-play (PnP) of distributed generators (DGs) in droop controlled microgrids, this paper proposes a preventive–corrective stability constrained optimal power flow (SC-OPF). Firstly, considering that DG droops are restricted by load sharing, we introduce the battery energy storage system (BESS) with adjustable droops, which acts as a new control resource. Secondly, we propose a bi-level optimization problem, where the upper level ensures minimizing the preventive–corrective coordinated cost and accommodating DG PnPs without violating stability constraints and the lower level focuses on the stability index given by semi-definite program (SDP) based on the Lyapunov equation. With the previously proposed analytical stability sensitivity, the lower level problem is reformulated by a series of linear constraints through the Benders decomposition method, and hence the entire bilevel problem can be solved efficiently and accurately. We apply the modified IEEE 33-bus system to validate the proposed SC-OPF model. From simulation results, the proposed model significantly improves the economic cost, stability, and computational efficiency compared to other SC-OPF models without considering BESS droop control or PnPs.

Identification and high-precision trajectory tracking control for space robotic manipulator

In this paper, an effective control and parameters identification scheme was proposed for high-precision trajectory tracking of space robotic manipulators. Our proposed control method employed a linear-extended-state-observer (LESO) based model predictive control (MPC) scheme. The feedback linearization approach was utilized to develop a model predictive controller. The optimization objective function was simplified to a standard quadratic programming (QP) problem for solution. To minimize the impact of the inaccuracy in the dynamics model, a LESO was designed to estimate the disturbance, and its convergence was demonstrated. In order to improve the accuracy of dynamic model, the physical consistency of inertial parameters was constrained using linear matrix inequality (LMI) techniques. Moreover, the dynamics parameters were estimated through semidefinite programming (SDP) techniques. To validate the effectiveness of the proposed control method, it was compared with existing controllers through simulations and experiments. The results demonstrated the superior performance of the proposed scheme, highlighting its potential for achieving high-precision trajectory tracking control in space robotic manipulators.

Quantum-enhanced greedy combinatorial optimization solver.

Combinatorial optimization is a broadly attractive area for potential quantum advantage, but no quantum algorithm has yet made the leap. Noise in quantum hardware remains a challenge, and more sophisticated quantum-classical algorithms are required to bolster their performance. Here, we introduce an iterative quantum heuristic optimization algorithm to solve combinatorial optimization problems. The quantum algorithm reduces to a classical greedy algorithm in the presence of strong noise. We implement the quantum algorithm on a programmable superconducting quantum system using up to 72 qubits for solving paradigmatic Sherrington-Kirkpatrick Ising spin glass problems. We find the quantum algorithm systematically outperforms its classical greedy counterpart, signaling a quantum enhancement. Moreover, we observe an absolute performance comparable with a state-of-the-art semidefinite programming method. Classical simulations of the algorithm illustrate that a key challenge to reaching quantum advantage remains improving the quantum device characteristics.

An Optimal Polarization SAR Three-Component Target Decomposition Based on Semi-Definite Programming

The model-based polarimetric synthetic aperture radar (PolSAR) target decomposition decodes the scattering mechanism of the target by analyzing the essential scattering components. This paper presents a new general three-component scattering power decomposition method by establishing optimization problems. It is known that the existing three-component decomposition method prioritizes the contribution of volume scattering, which often leads to volume scattering energy overestimation and may make double-bounce scattering and odd-bounce scattering component power negative. In this paper, a full parameter optimization method based on the remainder matrix is proposed, where all the elements of the coherency matrix will be taken into account including the remaining T13 component. The optimization is achieved with no priority order by solving the problem using semi-definite programming (SDP) based on the Schur complement theory. By doing so, the problem of volume scattering energy overestimation and negative powers will be avoided. The performance of the proposed approach is demonstrated and evaluated with AIRSAR and GF-3 PolSAR data sets. The experimental results show that by using the proposed method, the power contributions of volume scattering in two sets of data were reduced by at least 2.6% and 3.7% respectively, compared to traditional methods. And the appearance of negative power of double-bounce scattering and odd-bounce scattering are also avoided compared with those of the existing three-component decomposition.

Bootstrap, Markov Chain Monte Carlo, and LP/SDP hierarchy for the lattice Ising model

Bootstrap is an idea that imposing consistency conditions on a physical system may lead to rigorous and nontrivial statements about its physical observables. In this work, we discuss the bootstrap problem for the invariant measure of the stochastic Ising model defined as a Markov chain where probability bounds and invariance equations are imposed. It is described by a linear programming (LP) hierarchy whose asymptotic convergence is shown by explicitly constructing the invariant measure from the convergent sequence of moments. We also discuss the relation between the LP hierarchy for the invariant measure and a recently introduced semidefinite programming (SDP) hierarchy for the Gibbs measure of the statistical Ising model based on reflection positivity and spin-flip equations.

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 .

Strong Variational Sufficiency for Nonlinear Semidefinite Programming and Its Implications

Strong Variational Sufficiency for Nonlinear Semidefinite Programming and Its Implications