Abstract Achieving covariate balance in randomized experiments enhances the precision of treatment effect estimation. However, existing methods often require heuristic adjustments based on domain knowledge and are primarily developed for binary treatments. This paper presents Gaussianized Design Optimization, a novel framework for optimally balancing covariates in experimental design. The core idea is to Gaussianize the treatment assignments: we model treatments as transformations of random variables drawn from a multivariate Gaussian distribution and convert the design problem into a nonlinear continuous optimization over Gaussian covariance matrices. Compared to existing methods, our approach offers significant flexibility in optimizing covariate balance across a diverse range of designs and covariate types. Adapting the Burer–Monteiro approach for solving semidefinite programmes, we introduce first-order local algorithms for optimizing covariate balance, improving upon several widely used designs. Furthermore, we develop inferential procedures for constructing design-based confidence intervals under Gaussianization and extend the framework to accommodate continuous treatments. Simulations demonstrate the effectiveness of Gaussianization in multiple practical scenarios.

Many practically important NP-hard optimization problems are inherently higher-order polynomial optimizations, which are typically addressed using approximation algorithms. Classical relaxations express polynomial objectives over a polynomial basis and solve the resulting quadratic objective as a semidefinite program, which can significantly inflate problem size and degrade approximation behavior. Variational quantum analogues to classical semidefinite programs (vQSDPs) are near-term formulations geared towards quadratic objectives. We introduce Product-State Lifting (PSL), a simple product-register encoding that upgrades any vQSDP with basis-state encoding to tackle k -degree polynomial optimization. This upgrade requires only a linear increase in resources with constraints constant in k . As a worked example, we pair PSL with the recently-proposed vQSDP with the Hadamard test and approximate amplitude constraints [Quantum 7, 1057 (2023)], and outline an application to Max- k SAT. PSL maintains the device-friendly structure of vQSDPs while making polynomial degree a linear resource parameter, offering a general path from quadratic to polynomial optimization without the constraint growth typical of classical relaxations.

We show that, in many settings, the worst-case performance of a distributed optimization algorithm is independent of the number of agents in the system, and can thus be computed in the fundamental case with just two agents. This result relies on a novel approach that systematically exploits symmetries in worst-case performance computation, framed as Semidefinite Programming (SDP) via the Performance Estimation Problem (PEP) framework. Harnessing agent symmetries in the PEP yields compact problems whose size is independent of the number of agents in the system. When all agents are equivalent in the problem, we establish the explicit conditions under which the resulting worst-case performance is independent of the number of agents and is therefore equivalent to the basic case with two agents. Our compact PEP formulation also allows the consideration of multiple equivalence classes of agents, and its size only depends on the number of equivalence classes. This enables practical and automated performance analysis of distributed algorithms in numerous complex and realistic settings, such as the analysis of the worst agent performance. We leverage this new tool to analyze the performance of the EXTRA algorithm in advanced settings and its scalability with the number of agents, providing a tighter analysis and deeper understanding of the algorithm performance.

This paper proposes a Q-learning-based algorithm to solve the linear quadratic regulator (LQR) problem for unknown dynamic two-dimensional (2D) discrete-time systems. First, based on the value function formulation constructed using the Lyapunov function framework, algebraic Riccati inequality (ARI) and the Bellman inequality for solving the LQR problem are derived. Subsequently, a suboptimal state feedback controller is obtained based on these inequalities, and an offline policy iteration algorithm based on semi-definite programming (SDP) is introduced. On this foundation, by introducing the concept of Q-learning, the objective function and the Bellman inequality are transformed into the Q-function and its corresponding inequality. A Q-learning-based offline policy iteration equation is then derived, and further, an online policy iteration algorithm based on Q-learning is designed. Data are collected online during each iteration to solve the LQR problem for 2D discrete systems with unknown dynamics. Finally, the effectiveness of the proposed control scheme is validated through two examples.

Estimating spectral gaps of quantum many-body Hamiltonians is a highly challenging computational task, even under assumptions of locality and translation-invariance. Yet, the quest for rigorous gap certificates is motivated by their broad applicability, ranging from many-body physics to quantum computing and classical sampling techniques. Here we present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians in the thermodynamic limit. We formulate the gap certification problem as a hierarchy of optimization problems (semidefinite programs) in which the certificate – a proof of a lower bound on the gap – is improved with increasing levels. Our approach encompasses existing finite-size methods, such as Knabe's bound and its subsequent improvements, as those appear as particular possible solutions in our optimization, which is thus guaranteed to either match or surpass them. We demonstrate the power of the method on one-dimensional spin-chain models where we observe an improvement by several orders of magnitude over existing finite size criteria in both the accuracy of the lower bound on the gap, as well as the range of parameters in which a gap is detected.

Lateral stability preservation and disturbance rejection, such as crosswind resistance and unmodeled dynamics, remain critical challenges for autonomous vehicles (AVs) operating at handling limits during high-performance path tracking. This paper introduces a novel tube-based model predictive control (MPC) approach to enhance AV lateral stability and robustness. A composite strategy combining the modified lateral stability envelope (mLSE) and the equivalent Lyapunov stability-based yaw rate is proposed, effectively suppressing yaw rate oscillations while maintaining tracking accuracy. Furthermore, a semi-definite programming method is employed to derive a polyhedral robust positive invariant (pRPI) set and its associated feedback compensation measures, aiming to refine the mLSE and counteract disturbance effects. By integrating the mLSE, the equivalent yaw rate, and the pRPI set, the tube-based MPC controller is developed with mathematical proofs provided for the recursive feasibility and the input-to-state stability of the controller. Simulation experiments demonstrate that, in the presence of disturbances at handling limits, the proposed controller reduces yaw rate oscillations and improves tracking accuracy while maintaining robustness.

Semidefinite programming bounds and a Branch-and-bound algorithm for the Chordless Cycle Problem

Abstract This work deals with the variational determination of two-particle reduced density matrices corresponding to eigenstates of N-boson systems. Stringent N-representability conditions have been imposed in the variational treatment, in which the resulting optimization problem is addressed within a standard semidefinite programming framework. A unified variational treatment, based on the dispersion operator technique, is proposed. This treatment allows to determine ground-and excited-state energies and their corresponding reduced density matrices, which have been evaluated in systems formulated by means of simple Hamiltonians. We report results from different N-representability levels of treatment. These results are compared with those arising from the full configuration interaction procedure, what allows to point out the relevance of the imposed N-representability conditions. We also highlight the differences found in applying this methodology to fermion and boson systems.

A difference-of-convex approach for log-determinant semidefinite programming with chordal sparsity

This research introduces a novel convex methodology for estimating transmission loss coefficients (B-coefficients) in power systems using a data-driven approach based on power system measurements. To enhance estimation accuracy and practical relevance, the model is evaluated across a wide spectrum of operating conditions, incorporating random variations in active power injections and demand profiles modeled via uniform and Gaussian distributions. A semi-definite programming (SDP) model leveraging p-norm formulations is proposed to derive the B-coefficients efficiently. Numerical evaluations on IEEE 14-, 39-, 57-, and 118-bus test feeders demonstrate the effectiveness and robustness of the approach, yielding average estimation errors between and across diverse scenarios. These results confirm the reliability of the proposed methodology, contributing to improved accuracy in transmission loss modeling and supporting more efficient power system operations.

Internet of Things (IoT) is increasingly realized through large scale deployments of heterogeneous devices and gateways operating under strict energy budgets and interference limited links, which motivates reliability aware topology control and end to end communication performance objectives. As IoT deployments grow to massive scales and incorporate highly heterogeneous devices, designing and controlling network topology in a reliable and energy-efficient manner becomes a fundamental challenge. In particular, poor link quality, interference, and localization uncertainty severely limit the effectiveness of traditional topology-control approaches. In this paper, we address this challenge by introducing IoTNTop, a novel and unified graph-based framework for joint localization, graph embedding, and topology control in large-scale, resource-constrained IoT networks. Unlike conventional methods that decouple localization from topology design, IoTNTop embeds both end-nodes and gateways into a globally consistent spatial structure using partial and noisy distance measurements, and directly couples this geometry with communication-aware topology optimization. IoTNTop adopts an error-centric topology-control objective that explicitly minimizes end-to-end (E2E) error probability while enforcing practical code-rate and transmit-power constraints. The framework jointly optimizes link activation, transmit power, and data transmission code rate, and employs a scalable sub-graph stitching pipeline based on eigenvector synchronization (EVS), landmark alignment (LA), and semidefinite programming (SDP) refinement. A greedy signal-to-noise-ratio (SNR)-guided edge selection strategy with convergence checking further ensures computational efficiency. Comprehensive numerical analysis and network-level simulations show IoTNTop retains approximately 60-80% of the initial per-node energy budget while maintaining symbol error probability below 15% for the majority of nodes. At the same time, it converges in fewer iterations than Genetic Algorithm (GA) and brute-force baselines and sustains higher achievable code rates at lower transmit power levels. These performance gains remain consistent across the tested signal-to-noise ratio regimes and network sizes.

A reduced SQP-type algorithm for nonlinear semidefinite programming with LMI constraints

In this paper, we study a class of nonsmooth fractional programs (FP, for short) with SOS-convex semi-algebraic functions. Under suitable assumptions, we derive a strong duality result between the problem (FP) and its semidefinite programming (SDP) relaxations. Remarkably, we extract an optimal solution to the problem (FP) by solving one and only one associated SDP problem. Numerical examples are also given.

Recent works have proposed linear programming relaxations of variational optimization problems subject to nonlinear PDE constraints based on the occupation measure formalism. The main appeal of these methods is the fact that they rely on convex optimization, typically semidefinite programming. In this work we close an open question related to this approach. We prove that the classical and relaxed minima coincide when the dimension of the codomain of the unknown function equals one, both for calculus of variations and for optimal control problems, thereby complementing analogous results that existed for the case when the dimension of the domain equals one. In order to do so, we prove a generalization of the Hardt--Pitts decomposition of normal currents applicable in our setting, which is specific to the case where the dimension of the codomain equals one. We also show by means of a counterexample that, if both the dimensions of the domain and of the codomain are greater than one, there may be a positive gap. Finally, we show that in the presence of integral constraints, a positive gap may occur at any dimension of the domain and of the codomain.

Quantum state purification, a process that aims to recover a state closer to a system's principal eigenstate from multiple copies of an unknown noisy quantum state, is crucial for restoring noisy states to a more useful form in quantum information processing. Fault-tolerant quantum computation relies on stabilizer operations, which are classically simulable operations critical for error correction but inherently limited in computational power. In this Letter, we investigate the limitations of classically simulable operations for quantum state purification. We demonstrate that while certain classically simulable operations can enhance fidelity for specific noisy state ensembles, they cannot achieve universal purification. We prove that neither deterministic nor probabilistic protocols using only classically simulable operations can achieve universal two-to-one purification for qubit systems or systems of any odd dimension. We further extend this no-go result to state purification using three and four copies via semidefinite programming and numerical evidence. Our findings highlight the indispensable role of nonstabilizer resources and the inherent limitations of classically simulable operations in quantum state purification, emphasizing the necessity of harnessing the full power of quantum operations for robust quantum information processing.

The increasing penetration of renewable energy sources (RES) into modern power systems introduces small-signal synchronous stability challenges and constrains the power export capacity of renewable energy bases. To overcome these challenges, this paper proposes a two-stage coordinated dispatch strategy for energy storage and renewable generation, explicitly incorporating small-signal stability constraints. A generalized short-circuit ratio (gSCR) index is adopted to quantitatively evaluate the system's stability margin. To efficiently address the resulting nonlinear and non-convex optimization problem, a semidefinite programming (SDP)-based reformulation is developed, enabling tractable computation while ensuring stability compliance. The effectiveness and practical value of the proposed approach are demonstrated through detailed case studies on a realistic 7-bus renewable energy base system. Simulation results indicate that the proposed method increases the renewable energy export capacity by 7.79% while maintaining the required stability margins.

Abstract Quantum coherence, as one of the most fundamental non-classical features in quantum mechanics, plays a pivotal role in various quantum information processing tasks, including quantum computing and quantum metrology. The robustness of quantum coherence (RoC) offers an operational interpretation by quantifying the advantage provided by a quantum state in phase discrimination tasks. To achieve verification RoC with high precision via semidefinite programming (SDP), complete knowledge of quantum states is typically required. Relying solely on expectation values of observables may introduce significant errors in SDP-based estimations. To estimate RoC with high precision using limited data extracted from quantum states, firstly, we propose a semi-supervised K-nearest neighbor (KNN) algorithm (semi-KNN) and a semi-supervised method that combines the KNN and random forest (RF) models with a dynamical threshold (semi-KNN-RF). Then we implement the semi-KNN and semi-KNN-RF models to efficiently estimate quantum coherence by analyzing statistical data obtained from randomly generated local projective measurements performed on unknown quantum states. The semi-KNN-RF model performs better than the semi-KNN model. This innovative methodology allows for accurate coherence estimation, even for high-dimensional quantum systems.

Physics-informed data-driven modeling of AC optimal power flow via quadratic approximation

Abstract In this paper, we study a broad class of constrained best approximation problems in the face of data uncertainty. Adopting the deterministic robust optimisation framework, we model uncertainty using spectrahedral sets, which unify many convex uncertainty models commonly employed in practice. In this setting, the robust best approximation problem requires the solution to satisfy all constraints for every parameter in the prescribed spectrahedral sets, giving rise to a computationally difficult infinite-dimensional problem that may also involve infinitely many constraints. To address this difficulty, we employ convex optimisation duality and semi-definite optimisation techniques, and a variable transformation to reformulate the Lagrangian dual of the robust best approximation problem into a finite-dimensional convex semi-definite program. This enables the best approximation to be found from the solution of the dual problem via a solution recovery formula. By further reformulation of the dual to a convex composite unconstrained optimisation problem using convex conjugation techniques, we present a readily implementable first-order primal-dual proximal splitting method to compute the solution to the robust best approximation problem. Finally, we present the results of numerical experiments to illustrate our proposed approach.

Abstract Biclustering, also known as co-clustering or two-way clustering, simultaneously partitions the rows and columns of a data matrix to reveal submatrices with coherent patterns. Incorporating background knowledge into clustering to enhance solution quality and interpretability has attracted growing interest in mathematical optimization and machine learning research. Extending this paradigm to biclustering enables prior information to guide the joint grouping of rows and columns. We study constrained biclustering with pairwise constraints, namely must-link and cannot-link constraints, which specify whether objects should belong to the same or different biclusters. As a model problem, we address the constrained version of the k-densest disjoint biclique problem, which aims to identify k disjoint complete bipartite subgraphs (called bicliques) in a weighted complete bipartite graph, maximizing the total density while satisfying pairwise constraints. We propose both exact and heuristic algorithms. The exact approach is a tailored branch-and-cut algorithm based on a low-dimensional semidefinite programming (SDP) relaxation, strengthened with valid inequalities and solved in a cutting-plane fashion. Exploiting integer programming tools, a rounding scheme converts SDP solutions into feasible biclusterings at each node. For large-scale instances, we introduce an efficient heuristic based on the low-rank factorization of the SDP. The resulting nonlinear optimization problem is tackled with an augmented Lagrangian method, where the subproblem is solved by decomposition through a block-coordinate projected gradient algorithm. Extensive experiments on synthetic and real-world datasets show that the exact method significantly outperforms general-purpose solvers, while the heuristic achieves high-quality solutions efficiently on large instances.