Convex Optimization Research Articles

Excitation coefficients with a low dynamic range ratio (DRR) are advantageous in controlling mutual coupling between the elements of an antenna array. Their use also reduces the output power loss and simplifies the design of the feeding network. In this paper, a hybrid algorithm based on invasive weed optimization and convex optimization for the synthesis of distributed arrays with two subarrays is proposed. Arrays of this type are used in numerous applications, e.g. in aircraft. A constraint is added to the optimization problem to control the DRR of the array's excitation vector. Numerical results are presented for position-only, as well as for position and excitation control approaches. The trade-off between the peak sidelobe ratio and the obtained DRR is illustrated by numerical examples.

Abstract Integration of various electricity-generating technologies (such as natural gas, wind, nuclear, etc.) with storage systems (such as thermal, battery electric, hydrogen, etc.) has the potential to improve the economic competitiveness of modern energy systems. Driven by the need to efficiently assess the economic feasibility of various energy system configurations in early system concept development, this work outlines a versatile computational framework for assessing the net present value of various integrated storage technologies. The subsystems' fundamental dynamics are defined, with a particular emphasis on balancing critical physical and economic domains to enable optimal decision-making in the context of capacity and dispatch optimization. In its presented form, the framework formulates a linear, convex optimization problem that can be efficiently solved using a direct transcription approach in the open-source software DTQP. Three case studies demonstrate and validate the framework's capabilities, highlighting its value and computational efficiency in facilitating the economic assessment of various energy system configurations. In particular, natural gas with thermal storage and carbon capture, wind energy with battery storage, and nuclear with hydrogen are demonstrated.

ABSTRACT This article addresses the problem of synchronizing discrete‐time Lur'e type complex dynamic networks (CDNs) via dynamic event‐triggered control. In particular, it is considered that the control signal of each node is subject to input saturation. Using the Lyapunov Stability Theory, properties of slope‐restricted nonlinearities, and the linear matrix inequalities framework, constructive sufficient conditions are provided to ensure regional exponential synchronization of the CDN. Differently from other works in the current literature on CDNs, which require a trial‐and‐error procedure to select the event‐triggering mechanism (ETM) parameters, two systematic approaches, based on convex optimization, are presented to simultaneously synthesize (co‐design) the control law gains and the event‐generator parameters, aiming to reduce the number of events compared to a time‐triggered policy, with formal guarantees of regional synchronization with respect to a given admissible set of initial synchronization errors. Finally, a numerical example is presented to illustrate these approaches.

Optimal PI Disturbance Observer Design via Convex Optimization for Relative Degree-one Systems and Its Application to a Surface-mounted PMSM Back-EMF Estimator

With the increasing penetration of renewable energy generation, large-scale voltage dips may cause significant active power deficits and threaten system frequency stability. To address the issue, this article proposes a two-stage control strategy to enhance the frequency support capability of renewable energy plants by maximizing converter utilization during asymmetric grid voltage dips. First, a qualitative analysis of converter active power capacity considering current capacity constraints under grid faults is conducted to establish the basis for mitigating system-wide active power deficits. Second, individual phase current constraints are formulated for converters under asymmetric voltage conditions to achieve full utilization of converter capacity. Based on this, a two-stage control strategy for renewable energy plants is proposed, where plant-level convex optimization models for both pre-fault and post-fault conditions are established. By optimally allocating current references of converters within the plants, the requirement of grid codes is satisfied, and the overall frequency support capability of plants is effectively improved. Simulation results demonstrate that the proposed strategy raises the system frequency nadir from 49.58 Hz to 49.66 Hz under a minor fault and from 49.06 Hz to 49.11 Hz under a severe fault, confirming its effectiveness in enhancing the frequency support capability of renewable energy plants.

Efficient resist modeling and calibration using a Wiener–Padé formulation and convex optimizations

Continuum robots (CR) can achieve excellent dexterity and flexibility, making them suitable for navigating through cluttered environments and safely interacting with obstacles. Due to the underactuated nature of CRs, the contact mode between the robot and environment affects the static robot configuration. We show that the configuration space topology induced by environmental obstacles can be characterized by a quotient structure with a quotient space consisting of zero-actuation configurations. We propose to use the quotient space as a road map for motion planning to reduce computational load for exploration. Specifically, we propose an algorithm that identifies the quotient space as a graph of configuration modes by constructing a graph of convex sets in the free workspace, conducting tree search and convex optimizations to find candidate configurations, and then using elastic energy minimization to find the modes. We then use a motion planner which finds a path in the quotient space graph and constructs a continuous path in the configuration space. We demonstrate our method in several complex 3D environments and show that our method outperforms baselines in terms of computation time and success rate.

Explainable artificial intelligence based microscopic peripheral blood cell image classification by exploiting quadradic convex optimization

This paper develops the linear-algebraic foundations of the Convex Least Squares Programming (CLSP) estimator and constructs its modular two-step convex optimization framework, capable of addressing ill-posed and underdetermined problems. After reformulating a problem in its canonical form, A(r)z(r)=b, Step 1 yields an iterated (if r>1) minimum-norm least-squares estimate z^(r)=(AZ(r))†b on a constrained subspace defined by a symmetric idempotent Z (reducing to the Moore–Penrose pseudoinverse when Z=I). The optional Step 2 corrects z^(r) by solving a convex program, which penalizes deviations using a Lasso/Ridge/Elastic net-inspired scheme parameterized by α∈[0,1] and yields z^*. The second step guarantees a unique solution for α∈(0,1] and coincides with the Minimum-Norm BLUE (MNBLUE) when α=1. This paper also proposes an analysis of numerical stability and CLSP-specific goodness-of-fit statistics, such as partial R2, normalized RMSE (NRMSE), Monte Carlo t-tests for the mean of NRMSE, and condition-number-based confidence bands. The three special CLSP problem cases are then tested in a 50,000-iteration Monte Carlo experiment and on simulated numerical examples. The estimator has a wide range of applications, including interpolating input–output tables and structural matrices.

This paper presents a rigorous, application-oriented survey of separation theorems and supporting functionals for convex sets in normed and Banach spaces. Emphasizing geometric version of the Hahn-Banach theorem, this work develops clean conditions for strict and non-strict separation, existence of supporting hyperplanes, and links to dual cones and polar sets. We highlight the role of weak and weak-star topologies in separation and illustrate how these results undergird feasibility, sensitivity, and duality principles in convex optimization. Short, self-contained proofs and examples are provided to keep the exposition accessible while maintaining mathematical precision. The paper thereby complements classical treatments of linear functional extension by focusing on geometric separation mechanisms and their applied consequences, especially in linear programming, convex feasibility, and basic duality frameworks. This comprehensive investigation into separation theorems reveals fundamental geometric structures that underpin both theoretical functional analysis and practical optimization applications, providing a unified framework for understanding convex separation phenomena across diverse mathematical contexts.

Low-Rank Approaches to Large-Scale DNN Optimization Doubly nonnegative (DNN) relaxations are a powerful tool for approximating large-scale mixed-binary quadratic programs, but their size—often involving millions of constraints—makes them difficult to solve. We propose RiNNAL, a Riemannian augmented Lagrangian method that exploits the low-rank structure often present in optimal solutions. By applying a low-rank decomposition, RiNNAL reformulates most quadratic constraints into simpler affine ones, reducing problem complexity and mitigating issues caused by violations of Slater’s condition. A key innovation is showing that the required metric projection onto a certain algebraic variety, although nonconvex in form, can be solved as a convex optimization problem under mild regularity conditions, enforced through constraint reformulation. RiNNAL is versatile, handling not only DNN relaxations but also general semidefinite programs with polyhedral constraints. Extensive experiments on challenging benchmarks confirm its efficiency and robustness, making RiNNAL a promising solver for large-scale optimization problems.

ABSTRACT We describe the construction of dual variational principles and action functionals for nonlinear dynamical systems using a methodology based on the dual Lagrange multiplier formalism and a convex optimization approach, to derive families of dual actions that correspond to the given nonlinear ordinary differential system. The method involves a mapping between the original or primal variables and their duals, with the primal variables being expressible in terms of the dual variables and their derivatives and will be referred to as the dual‐to‐primary (or primal) (DtP) map. We demonstrate the application of this technique to the three‐dimensional, autonomous Chen–Lee system, known for its chaotic behavior, and the Hopf–Langford system, which arises in the context of turbulence studies. The derived dual action principles provide an alternative variational description of these nonlinear systems in terms of the Euler–Lagrange equations.

ABSTRACT This article considers the probabilistic‐constrained tracking problem of a networked nonlinear system subject to hybrid attacks on the sensor and actuator channels. Two independent Markov‐based attack models are proposed for characterizing a mixture of denial‐of‐service, deception and replay attacks. We generalize the attack models in the sense that Markov processes are more practical and complicated than Bernoulli ones, and the attacks at the actuator end are also considered, which prevents the control input from reaching the actuator. A resilient receding‐horizon control law is designed based on the seriously tampered output, consisting of a probability‐based observer and a convex optimization procedure for control parameters. It is capable of mitigating the attacks on both channels and fulfilling the tracking tasks by establishing a trade‐off between the volume of the constrained set and the violation probability of the tracking error.

Convex Optimization plays a crucial role in various scientific and industrial applications, such as economics, engineering, and computer science, with a primary focus on linear and quadratic optimization. This study examines the characteristics and comparison between linear and quadratic optimization, two main subclasses of convex optimization. Linear optimization (LP) is characterized by a linear objective function and linear constraints, where classical methods such as Simplex and Interior-Point are used for efficient solutions. In contrast, quadratic optimization (QP) involves a convex quadratic objective function with linear constraints, requiring more complex methods such as KKT factorization, Schur-Complement, Null-Space, Active-Set, and Interior-Point for solving. This paper summarizes various solution methods for both types of optimizations and compares their strengths and limitations. The key findings indicate that linear optimization is simpler and more efficient, while quadratic optimization offers greater flexibility in modeling problems with more complex structures. The study concludes that a deep understanding of both approaches is essential for the development of more efficient and applicable convex optimization algorithms.

On Differential Stability of a Class of Convex Optimization Problems

A Systematic Approach to General Higher-Order Majorization-Minimization Algorithms for (Non)convex Optimization

An overarching objective in contemporary statistical network analysis is extracting salient information from datasets consisting of multiple networks. To date, considerable attention has been devoted to node and network clustering, while comparatively less attention has been devoted to downstream connectivity estimation and parsimonious embedding dimension selection. Given a sample of potentially heterogeneous networks, this article proposes a method to simultaneously estimate a latent matrix of connectivity probabilities and its embedding dimensionality or rank after first pre-estimating the number of communities and the node community memberships. The method is formulated as a convex optimization problem and solved using an alternating direction method of multipliers algorithm. We establish estimation error bounds under the Frobenius norm and nuclear norm for settings in which observable networks have blockmodel structure, even when node memberships are imperfectly recovered. When perfect membership recovery is possible and dimensionality is much smaller than the number of communities, the proposed method outperforms conventional averaging-based methods for estimating connectivity and dimensionality. Numerical studies empirically demonstrate the accuracy of our method across various scenarios. Additionally, analysis of a primate brain dataset demonstrates that posited connectivity is not necessarily full rank in practice, illustrating the need for flexible methodology.

Abstract We introduce a computer-assisted proof for the required number of electrodes for uniqueness and global reconstruction for the inverse Robin transmission problem, where the corrosion function on the boundary of an interior object is to be determined from electrode current-voltage measurements. We consider the shunt electrode model where, in contrast to the standard Neumann boundary condition, the applied electrical current is only partially known. The aim is to determine the corrosion coefficient with a finite number of measurements.

In this paper, we present a numerically verifiable criterion that ensures unique solvability of the inverse problem, given a desired resolution. This allows us to explicitly determine the required number and position of the electrodes. Furthermore, we will present an error estimate for noisy data. By rewriting the problem as a convex optimization problem, our aim is to develop a globally convergent reconstruction algorithm.

The rapid growth of Industrial Internet of Things (IIoT) terminals has resulted in tasks exhibiting increased concurrency, heterogeneous resource demands, and dynamic priorities, significantly increasing the complexity of task scheduling in edge computing. Cloud-edge-end collaborative computing leverages cross-layer task offloading to alleviate edge node resource contention and improve task scheduling efficiency. However, existing methods generally neglect the joint optimization of task offloading, resource allocation, and priority adaptation, making it difficult to balance task execution and resource utilization under resource-constrained and competitive conditions. To address this, this paper proposes a two-stage dynamic-priority-aware joint task offloading and resource allocation method (DPTORA). In the first stage, an improved Multi-Agent Proximal Policy Optimization (MAPPO) algorithm integrated with a Priority-Gated Attention Module (PGAM) enhances the robustness and accuracy of offloading strategies under dynamic priorities; in the second stage, the resource allocation problem is formulated as a single-objective convex optimization task and solved globally using the Lagrangian dual method. Simulation results show that DPTORA significantly outperforms existing multi-agent reinforcement learning baselines in terms of task latency, energy consumption, and the task completion rate.

This paper analyzes the channel estimation of rate splitting multiple access (RSMA) wireless network through the full-duplex amplify-and-forward (AF) relay. Basically, full-duplex transmission can improve temporal efficiency, however the loop interference is an unavoidable problem that occurs in the strong user of this proposed network. The orthogonal frequency division multiplexing (OFDM) system is used to provide high data rate communication, assuming the presence of phase noise (PN) in local oscillators. Using the least square (LS) estimate, the channel coefficients of the proposed RSMA relay network are estimated. In addition, convex optimization techniques are applied to estimate the phase noise components of this network. The problem is formulated by optimizing phase noise under transmit power constraints. We analyze the Bit Error Rate (BER) performance of the proposed network under binary phase shift keying (BPSK) modulation and 16-quadrature amplitude modulation (QAM). Simulation results demonstrate that channel estimation achieves better performance after the PN compensation.