Semidefinite Programming Research Articles

Automated tight Lyapunov analysis for first-order methods

AbstractWe present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider (i) classes of optimization problems of finite-sum form with (possibly strongly) convex and possibly smooth functional components, (ii) first-order methods that can be written as a linear system on state-space form in feedback interconnection with the subdifferentials of the functional components of the objective function, and (iii) quadratic Lyapunov inequalities that can be used to draw convergence conclusions. We present a necessary and sufficient condition for the existence of a quadratic Lyapunov inequality within a predefined class of Lyapunov inequalities, which amounts to solving a small-sized semidefinite program. We showcase our methodology on several first-order methods that fit the framework. Most notably, our methodology allows us to significantly extend the region of parameter choices that allow for duality gap convergence in the Chambolle–Pock method when the linear operator is the identity mapping.

3D Genome Reconstruction from Partially Phased Hi-C Data

The 3-dimensional (3D) structure of the genome is of significant importance for many cellular processes. In this paper, we study the problem of reconstructing the 3D structure of chromosomes from Hi-C data of diploid organisms, which poses additional challenges compared to the better-studied haploid setting. With the help of techniques from algebraic geometry, we prove that a small amount of phased data is sufficient to ensure finite identifiability, both for noiseless and noisy data. In the light of these results, we propose a new 3D reconstruction method based on semidefinite programming, paired with numerical algebraic geometry and local optimization. The performance of this method is tested on several simulated datasets under different noise levels and with different amounts of phased data. We also apply it to a real dataset from mouse X chromosomes, and we are then able to recover previously known structural features.

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.

A distributionally robust chance-constrained kernel-free quadratic surface support vector machine

This paper studies the problem of constructing a robust nonlinear classifier when the data set involves uncertainty and only the first- and second-order moments are known a priori. A distributionally robust chance-constrained kernel-free quadratic surface support vector machine (SVM) model is proposed using the moment information of the uncertain data. The proposed model is reformulated as a semidefinite programming problem and a second-order cone programming problem for efficient computations. A geometric interpretation of the proposed model is also provided. For commonly used data without prescribed uncertainty, a cluster-based data-driven approach is introduced to retrieve the hidden moment information that enables the proposed model for robust classification. Extensive computational experiments using synthetic and public benchmark data sets with or without uncertainty involved support the superior performance of the proposed model over other state-of-the-art SVM models, particularly when the data sets are massive and/or imbalanced.

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

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

Semi-definite programming and quantum information

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

IPRSDP: a primal-dual interior-point relaxation algorithm for semidefinite programming

IPRSDP: a primal-dual interior-point relaxation algorithm for semidefinite programming

Numerical investigation of effective nonlinear coefficient model for coupled third harmonic generation

In this paper, the optimal solution of effective nonlinear coefficient of quasi-phase-matching (QPM) crystals for coupled third harmonic generation (CTHG) was numerically investigated. The effective nonlinear coefficient of CTHG was converted to an Ising model for optimizing domain distributions of aperiodically poled lithium niobate (APPLN) crystals with lengths as 0.5 mm and 1 mm, and fundamental wavelengths ranging from 1000 nm to 6000 nm. A method for reconstructing crystal domain poling weight curve of coupled nonlinear processes was also proposed, which demonstrated the optimal conversion ratio between two coupled nonlinear processes at each place along the crystal. In addition, by applying the semidefinite programming, the upper bound on the effective nonlinear coefficients deff for different fundamental wavelengths were calculated. The research can be extended to any coupled dual χ(2) process and will help us to understand better the dynamics of coupled nonlinear interactions based on QPM crystals.

Structured singular value of a repeated complex full‐block uncertainty

AbstractThe structured singular value (SSV), or , is used to assess the robust stability and performance of an uncertain linear time‐invariant system. Existing algorithms compute upper and lower bounds on the SSV for structured uncertainties that contain repeated (real or complex) scalars and/or nonrepeated complex full‐blocks. This paper presents algorithms to compute bounds on the SSV for the case of repeated complex full‐blocks. This specific class of uncertainty is relevant for the input‐output analysis of many convective systems, such as fluid flows. Specifically, we present a power iteration to compute the SSV lower bound for the case of repeated complex full‐blocks. This generalizes existing power iterations for repeated complex scalars and nonrepeated complex full‐blocks. The upper bound can be formulated as a semi‐definite program (SDP), which we solve using a standard interior‐point method to compute optimal scaling matrices associated with the repeated full‐blocks. Our implementation of the method only requires gradient information, which improves the computational efficiency of the method. Finally, we test our proposed algorithms on an example model of incompressible fluid flow. The proposed methods provide less conservative bounds as compared to prior results, which ignore the repeated full‐block structure.

Gridless compressed covariance sensing for parameter identification of nonuniform and undersampled blade tip timing data

Due to the harsh working environments, rotating blades are susceptible to failures, which endanger the operational safety of turbomachinery; thus, it is important to monitor rotating blades to promptly detect incipient failure. Blade tip timing (BTT) is a promising approach for monitoring the health of rotating blades owing to its noncontact nature, high measurement efficiency, and long service life. However, the large gap between the high vibration frequency and low sampling frequency results in severe undersampling of the BTT signal. Although advanced methods have been proposed to reconstruct the spectrum or identify parameters from BTT signal, almost all of them face basis mismatch issue caused by frequency discretization and suffers from inevitable reconstruction errors. To overcome these limitations, we developed a gridless compressed covariance sensing (GCCS) approach for the sparse spectral estimation and parameter identification of BTT data. Unlike existing methods relying on frequency discretization, GCCS directly performs a continuous range of spectrum reconstruction and parameter estimation. We derived that GCCS is equivalent to a weighted atomic norm minimization (ANM) with the weight function linked to the Capon spectrum, which implies the advantages of GCCS over methods based on conventional ANM. GCCS involves two core steps, first, covariance (matrix) recovery from compressive samples is formulated as a semidefinite programming; second, the extended Prony’s method is employed as a postprocessing technique to retrieve vibration parameters from the covariance estimations, thereby achieving continuous parameter estimation. The quantitative comparison in the simulations and experiments shows the effectiveness of GCCS in overcoming basis mismatch and improving the accuracy and consistency of the parameter estimation.

Synthesis of ship systems optimal regulators based on matrix inequalities

The output and state regulators synthesis for ship control systems based on linear matrix inequalities using the CVX software package for solving optimization problems by the method of semi-definite programming is considered. Estimates parameters for regulators providing asymptotically stable systems regimes on Lyapunov inequalities solutions sets are obtained. An algorithm for controlling the ship course using a robust regulator in state feedback, followed by the use of CVX technologies to implement the minimum energy consumption control algorithm, is presented. The control vector estimation is reduced to the minimization of the L2 norm and it is performed in two stages, namely, the best mode is first determined, and then the mode under constraints is defined. The use of Lyapunov inequalities with the calculations implementation in CVX format on proprietary solvers has allowed us to obtain compact program texts and expand the set of restrictions introduced. In the algorithm for solving the boundary value problem of controlling a discrete model of an object, in contrast to existing optimization methods, a mode of transfer from the initial state to the final one with the right boundary, not necessarily equal to zero, is implemented. For a class of observable and controllable time–invariant systems, using the Krylov’s function contained in the MATLAB gallery of functions, L2‑norms to be minimized are formed. Examples of regulator calculations with a given degree of stability and optimal systems, confirming proposed solutions correctness, are given.

Physical Layer Security of the MIMO-NOMA Systems under Near-Field Scenario

In this paper, we propose a secure transmission framework for near-field MIMO-NOMA systems. This architecture integrates beamforming mechanisms for both transmission and reception, allowing the base station to send encrypted information to authorized users, effectively countering eavesdropping attempts in a near-field environment. To optimize the secrecy communication capability in the near field, a two-phase alternating optimization algorithm is introduced. In the first phase, the semidefinite relaxation (SDR) method is used to relax constraints in the problem and convert it into a semidefinite programming (SDP) problem. In the second phase, the successive convex approximation (SCA) algorithm is employed to transform the original non-convex problem into a convex optimization problem, obtaining a locally optimal solution through multiple iterations. Simulation results validate that the proposed near-field communication strategy exhibits superior secrecy communication capabilities under various parameter settings compared to far-field communication strategies.

Virtual Quantum Resource Distillation.

Distillation, or purification, is central to the practical use of quantum resources in noisy settings often encountered in quantum communication and computation. Conventionally, distillation requires using some restricted "free" operations to convert a noisy state into one that approximates a desired pure state. Here, we propose to relax this setting by only requiring the approximation of the measurement statistics of a target pure state, which allows for additional classical postprocessing of the measurement outcomes. We show that this extended scenario, which we call "virtual resource distillation," provides considerable advantages over standard notions of distillation, allowing for the purification of noisy states from which no resources can be distilled conventionally. We show that general states can be virtually distilled with a cost (measurement overhead) that is inversely proportional to the amount of existing resource, and we develop methods to efficiently estimate such cost via convex and semidefinite programming, giving several computable bounds. We consider applications to coherence, entanglement, and magic distillation, and an explicit example in quantum teleportation (distributed quantum computing). This work opens a new avenue for investigating generalized ways to manipulate quantum resources.

Control/physical systems co-design with spectral temporal logic specifications and its applications to MEMS

‘Co-design’ problems try to simultaneously design the physical and control components to improve the overall system performance. However, existing co-design paradigms cannot deal with complex frequency temporal domain specifications. In this paper, we investigate the co-design problem for a class of linear parameter-varying (LPV) systems with frequency temporal domain specifications. Firstly, the frequency temporal domain specifications are written in a formal language called spectral temporal logic (STL). Secondly, the satisfaction conditions of the spectral temporal logic specifications have been transformed into non-linear matrices inequality forms with necessary and sufficient conditions. Thirdly, the co-design problem is transformed into a non-convex optimisation problem with mixed-integer linear matrix inequalities (MILMIs) constraints, and then an iterative algorithm is proposed to solve the co-design problem with semidefinite programming (SDP). Finally, the performance of the algorithm and the expressiveness of spectral temporal logic are illustrated with the applications to micro-electromechanical systems (MEMS).

Automorphic spectra and the conformal bootstrap

We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of representation theory of P S L 2 ( R ) \mathrm {PSL}_2(\mathbb {R}) and semidefinite programming, the method yields rigorous upper bounds on the Laplacian spectral gap. In several examples, the bound is nearly sharp. For instance, our bound on all genus-2 surfaces is λ 1 ≤ 3.8388976481 \lambda _1\leq 3.8388976481 , while the Bolza surface has λ 1 ≈ 3.838887258 \lambda _1\approx 3.838887258 . The bounds also allow us to determine the set of spectral gaps attained by all hyperbolic 2-orbifolds. Our methods can be generalized to higher-dimensional hyperbolic manifolds and to yield stronger bounds in the two-dimensional case. The ideas were closely inspired by modern conformal bootstrap.

Effective field theory bootstrap, large-N χPT and holographic QCD

We review the effective field theory (EFT) bootstrap by formulating it as an infinite-dimensional semidefinite program (SDP), built from the crossing symmetric sum rules and the S-matrix primal ansatz. We apply the program to study the large-N chiral perturbation theory (χPT) and observe excellent convergence of EFT bounds between the dual (rule-out) and primal (rule-in) methods. This convergence aligns with the predictions of duality theory in SDP, enabling us to analyze the bound states and resonances in the ultra-violet (UV) spectrum. Furthermore, we incorporate the upper bound of unitarity to uniformly constrain the EFT space from the UV scale M using the primal method, thereby confirming the consistency of the large-N expansion. In the end, we translate the large-N χPT bounds to constrain the higher derivative corrections of holographic QCD models.

Using hierarchical information-theoretic criteria to optimize subsampling of extensive datasets

This paper addresses the challenge of subsampling large datasets, aiming to generate a smaller dataset that retains a significant portion of the original information. To achieve this objective, we present a subsampling algorithm that integrates hierarchical data partitioning with a specialized tool tailored to identify the most informative observations within a dataset for a specified underlying linear model, not necessarily first-order, relating responses and inputs. The hierarchical data partitioning procedure systematically and incrementally aggregates information from smaller-sized samples into new samples. Simultaneously, our selection tool employs Semidefinite Programming for numerical optimization to maximize the information content of the chosen observations. We validate the effectiveness of our algorithm through extensive testing, using both benchmark and real-world datasets. The real-world dataset is related to the physicochemical characterization of white variants of Portuguese Vinho Verde. Our results are highly promising, demonstrating the algorithm’s capability to efficiently identify and select the most informative observations while keeping computational requirements at a manageable level.

Control Optimization of the Offshore HVDC Grid based on Modular Multilevel Converter for Improving DC Voltage Stability

The aim of this paper is to reduce the voltage oscillation of the HVDC cable in an offshore point-to-point configuration based on the modular multilevel converter. An optimization methodology based on semidefinite programming is proposed, which improves the DC voltage stability under the worst-case initial perturbation scenario. This goal is achieved by implementing a centralized optimal linear feedback controller, which guarantees Lyapunov stability margins while considering constraints on control inputs and state variables. The scope of this work is twofold. First, to analyze small-signal eigenvalue stability of the modular multilevel converter-based HVDC grid to identify the source of the oscillations. Second, to use this analysis as a basis to design a centralized optimal linear feedback controller to improve stability margins. The linear feedback controller is first applied as an additional control layer to the standard control of the offshore point-to-point HVDC grid, and then it is implemented to substitute the droop control function of the grid. Finally, the applicability and validity of the results are verified by time-domain simulations.

Solving Sparse Separable Bilinear Programs Using Lifted Bilinear Cover Inequalities

Recently a class of second-order cone representable convex inequalities called lifted bilinear cover inequalities were introduced, which are valid for a set described by a separable bilinear constraint together with bounds on variables. In this paper, we study the computational potential of these inequalities for separable bilinear optimization problems. We first prove that the semidefinite programming relaxation provides no benefit over the McCormick relaxation for such problems. We then design a simple randomized separation heuristic for lifted bilinear cover inequalities. In our computational experiments, we separate many rounds of these inequalities starting from McCormick’s relaxation of instances where each constraint is a separable bilinear constraint set. We demonstrate that there is a significant improvement in the performance of a state-of-the-art global solver in terms of gap closed, when these inequalities are added at the root node compared with when they are not. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: S. S. Dey gratefully acknowledges the support by the Office of Naval Research [Grant N000141912323].

Jointly Active/Passive Beamforming Optimization for Intelligent-Reflecting Surface-Assisted Cognitive-IoT Networks

To overcome challenges such as limited energy availability for terminal devices, constrained network coverage, and suboptimal spectrum resource utilization, with the overarching objective of establishing a sustainable and efficient interconnection infrastructure, we introduce an innovative Intelligent Reflective Surface (IRS) technology. This cutting-edge IRS technology is employed to architect a wireless and energy-efficient cognitive secure communication network assisted by IRS. To further optimize the overall energy harvesting of this network, we present a cognitive secure resource allocation scheme, aiming to maximize the system’s total collected energy. This scheme carefully considers various constraints, including transmission power constraints for cognitive base stations, power constraints for jammer devices, interference limitations for all primary users, minimum security rate constraints for all cognitive Internet of Things (IoT) devices, and phase shift constraints for IRS. We establish a comprehensive hybrid cognitive secure resource allocation model, encompassing joint cognitive transmission beam design, jammer device transmission beam design, and phase shift design. Given the non-convex nature of the formulated problem and the intricate coupling relationships among variables, we devise an effective block coordinate descent (BCD) iterative algorithm. The realization of joint cognitive/jammer base station transmission beam design and phase shift design employs sophisticated techniques such as continuous convex approximation methods and semi-definite programming. Simulation results underscore the superior performance of the proposed scheme compared to existing resource allocation approaches, particularly in terms of total harvested energy and other critical metrics.