Convex Technique Research Articles

The circle packing problem: A theoretical comparison of various convexification techniques

We consider the problem of packing a given number of congruent circles with the maximum radius in a unit square as a mathematical optimization problem. Due to the presence of non-overlapping constraints, this problem is a notoriously difficult nonconvex quadratically constrained optimization problem, which possesses many local optima. We consider several popular convexification techniques, giving rise to linear programming relaxations and semidefinite programming relaxations for the circle packing problem. We compare the strength of these relaxations theoretically, thereby proving the conjectures by Anstreicher. Our results serve as a theoretical justification for the ineffectiveness of existing machinery for convexification of non-overlapping constraints.

Optimal Control of a Roll-to-Roll Dry Transfer Process With Bounded Dynamics Convexification

Abstract For efficient roll-to-roll (R2R) production of flexible electronic components, a precise R2R transfer peeling process is essential, requiring accurate modeling and control. This paper introduces a novel approach to confining the dynamics of a nonlinear R2R mechanical peeling system within a convex set known as a norm-bounded linear differential inclusion (NLDI). This method utilizes constraints on uncertain system variables to create a tighter NLDI representation compared to other convexification techniques. Moreover, it offers drastically reduced computational cost compared to previous methods applied to convexify the R2R peeling system. The NLDI is employed to generate an H∞-optimal controller for the R2R peeling system, and both simulations and experiments demonstrate better dynamic performance compared to other controllers for R2R transfer.

Convexification techniques for fractional programs

Convexification techniques for fractional programs

Chance constrained optimal power-water flow: An iterative algorithm assisted by deep neural networks

The optimal power-water flow (OPWF) is a fundamental tool for operating integrated power and water networks. To address uncertainties in renewable energy and load demands, researchers have used chance-constrained models to investigate OPWF under uncertainty. However, existing work has limitations, such as oversimplified network representations, intractable problem formulations, and high computational burdens. This paper presents a chance-constrained OPWF formulation that incorporates detailed network constraints and stochastic uncertainties from wind speeds and electricity/water demands. By leveraging the concept of quantiles of stochastic variables and convexification techniques, the model is recast into a deterministic mixed-integer second-order cone programming (MISOCP) problem with uncertain margins. To efficiently solve this problem, we propose a novel iterative algorithm assisted by deep neural networks (DNNs). The algorithm integrates two DNNs: one designed to solve power and water flow equations, accelerating the probabilistic evaluation of uncertain margins; and the other to predict pipeline flows and refine variable ranges, facilitating a warm-start approach to the MISOCP model. Case studies conducted on the IEEE 123-node feeder coupled with a 67-node water distribution network validate the two DNNs and the overall iterative algorithm. The findings demonstrate that the proposed DNN-assisted approach effectively balances system security and economy. Compared to existing algorithms, the relative error of the optimization solution is less than 0.03%, with a speedup ratio exceeding 200.

Finite frequency H ∞ control for singular systems with application to a 2-machine 6-bus power network

This paper is concerned with the H ∞ control problem for singular systems over finite frequency ranges. The disturbance is assumed to be within low-/middle-/high-frequency ranges. Both the state feedback and static output feedback controllers are designed. First, by combining the Lyapunov function method and the generalized Kalman–Yakubovich–Popov lemma, a new finite frequency bounded real lemma (BRL) is proposed, which ensures the resulting closed-loop system to be regular, impulse free and stable with given finite frequency H ∞ performance bound. It is shown by theoretical analysis and numerical simulation that the proposed BRL has less conservatism than the existing ones in the literature. With the aid of the Finsler's Lemma, two improved finite frequency BRLs are further presented for the design of controllers. Then, by utilizing convexification techniques, sufficient conditions for solving the desired state feedback and static output feedback gains are presented in terms of linear matrix inequalities, respectively. Finally, the applicability and effectiveness of the proposed control methods are verified by a 6-bus power network system.

Aperiodic sampled‐data stabilization of switched singular systems with asynchronous switching: A hybrid control approach

AbstractThis article investigates the sampled‐data stabilization problem for a class of switched singular systems with aperiodic sampling. The asynchronous switching between the plant and the switched controller caused by sampling is considered. Unlike normal systems, the conventional zero‐order hold type sampled‐data control law may be inapplicable to singular systems. This is because under that control law, the algebraic equations of singular systems may have no solutions at sampling instants. To cope with this problem, a new hybrid control law composed of a mode‐dependent impulsive controller and a mode‐dependent sampled‐data state feedback controller is devised. The impulsive controller adjusts the values of differential substate at sampling instants so that the algebraic equations are solvable. Then, by constructing a novel piecewise sampling‐time‐dependent Lyapunov function and using the average dwell time method as well as convex technique, a sampling‐range‐dependent sufficient condition for exponential stability of the resulting closed‐loop system is established. Moreover, an optimization‐based method is proposed to solve the hybrid controller gains. Simulation results on two examples including a mechanical arm system show the effectiveness of the proposed control method.

Construction of 4 x 4 symmetric stochastic matrices with given spectra

Abstract The symmetric stochastic inverse eigenvalue problem (SSIEP) asks which lists of real numbers occur as the spectra of symmetric stochastic matrices. When the cardinality of a list is 4, Kaddoura and Mourad provided a sufficient condition for SSIEP by a mapping and convexity technique. They also conjectured that the sufficient condition is the necessary condition. This study presents the same sufficient condition for SSIEP, but we do it in terms of the list elements. In this way, we provide a different but more straightforward construction of symmetric stochastic matrices for SSIEP compared to those of Kaddoura and Mourad.

Generalized Nash equilibrium problems with mixed-integer variables

AbstractWe consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido–Isoda function. To any given instance of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.

Analysis of finite-time stability in genetic regulatory networks with interval time-varying delays and leakage delay effects

<p>We primarily examined the effect of leakage delays on finite-time stability problems for genetic regulatory networks with interval time-varying delays. Since leakage delays can occur within the negative feedback components of networks and significantly impact their dynamics, they may potentially cause instability or suboptimal performance. The derived criteria encompass both leakage delays and discrete interval time-varying delays through the construction of a Lyapunov-Krasovskii function. We employed the estimation of various integral inequalities and a reciprocally convex technique. Additionally, these models consider lower bounds on delays, which may be either positive or zero, and allow for the derivatives of delays to be either positive or negative. Consequently, new criteria for genetic regulatory networks with interval time-varying delays under the effect of leakage delays are expressed in the form of linear matrix inequalities. Ultimately, a numerical example is presented to show the effect of leakage delays and to emphasize the significance of our theoretical findings.</p>

The backward problem for time-fractional evolution equations

In this paper, we consider the backward problem for fractional in time evolution equations ∂ t α u ( t ) = Au ( t ) with the Caputo derivative of order 0 < α ≤ 1 , where A is a self-adjoint and bounded above operator on a Hilbert space H. First, we extend the logarithmic convexity technique to the fractional framework by analyzing the properties of the Mittag–Leffler functions. Then we prove conditional stability estimates of Hölder type for initial conditions under a weaker norm of the final data. Finally, we give several applications to show the applicability of our abstract results.

New Results on Nonsynchronous-Observer-Based Output-Feedback Control of Fuzzy-Affine-Model-Based Discrete-Time Nonlinear Systems

This work investigates the non-synchronous observer-based output feedback control of nonlinear systems by T-S fuzzy affine models. In order to estimate the unmeasurable system states and handle the non-synchronization issue of premise variables between the observer/controller and the original system, a new non-synchronous piecewise fuzzy affine observer is firstly synthesized to estimate the unmeasurable system state variables. Via utilizing some elegant matrix inequalities with convexification techniques, new observer-based piecewise output feedback controller design results are proposed within the framework of piecewise quadratic Lyapunov functions. Simulation studies are given to illustrate the validity of the developed approach.

Robust Preview Path Tracking Control of Autonomous Vehicles Under Time-Varying System Delays and Saturation

Realistic uncertainties and disturbances cannot be ignored when striving towards high-performance autonomous driving. Meanwhile, advanced vehicular sensing technology enables more and more self-driving techniques to come out. Against this backdrop, this paper aims to holistically address the system uncertainties and disturbances in the path tracking control of autonomous vehicles (AVs) by the preview control theory to achieve superior control performance. First, unmeasureable multi-uncertainties and the external disturbance are considered simultaneously into the robust gain-scheduling framework with variable longitudinal velocity. Then, by the equivalent delay approach, ineluctable time-varying system delays are effectively handled. Next, the physical constraints of the steering system and direct yaw moment control (DYC) system are considered for practical implementation through robust set invariance theory. The parallel distributed compensation (PDC) technique and auxiliary feedback matrices approach are introduced for reducing design conservatism. After that, by incorporating the road curvature into the augmented system state, the path tracking preview control problem can be solved without online optimization. Thereafter, the closed-loop system stability and guaranteed performance have been proved through the Lyapunov- Krasovski approach. The proposed controller design is finally reformulated as the optimization problem based on linear matrix inequalities (LMIs) via specific convexification techniques. Finally, to verify the validity of the proposed controller, Software in Loop (SIL) tests with Carsim full-vehicle model under different dynamic conditions have been implemented.

Fuzzy-Affine-Model-Based Filtering Design for Continuous-Time Roesser-Type 2-D Nonlinear Systems.

This article tackles the problem of filtering design for continuous-time Roesser-type 2-D nonlinear systems via Takagi-Sugeno (T-S) fuzzy affine models. The aim is to design an admissible piecewise affine (PWA) filter such that the filtering error system is asymptotically stable with a prescribed disturbance attenuation level. First, 2-D Roesser nonlinear systems are approximated by a kind of 2-D fuzzy affine models with norm-bounded uncertainties. Then, the premise variable space of the 2-D fuzzy affine systems is partitioned into two classes of subspaces, that is: 1) crisp regions and 2) fuzzy regions. For each region, boundary continuity matrices and characterizing matrices are constructed by utilizing the space partition information and 2-D structure. After that, novel piecewise Lyapunov functions are constructed, based on which together with S-procedure, the asymptotic stability with performance is guaranteed for the filtering error system. By the projection lemma and some elegant convexification techniques, the PWA filtering design conditions are obtained. Finally, the less conservativeness and effectiveness of the proposed approach over a common Lyapunov function-based one are illustrated by simulation studies.

Fuzzy Observer-Based Output Feedback Control of Continuous-Time Nonlinear Two-Dimensional Systems

The fuzzy observer-based output feedback control of continuous-time nonlinear two-dimensional systems is studied by T-S fuzzy models in this work. The plant information of the two-dimensional systems evolves along two independent directions dynamically. The nonlinear two-dimensional systems are firstly expressed by Takagi-Sugeno fuzzy models with parameter uncertainties. By a Lyapunov method together with some convexification techniques, two methods are developed for fuzzy observer-based output feedback controller synthesis of the underlying fuzzy two-dimensional systems, and novel output feedback controller synthesis results are proposed within a convex optimization setup. Simulation studies are provided to illustrate the validity of the proposed methods.

Optimal Placement and Sizing of D-STATCOMs in Electrical Distribution Networks Using a Stochastic Mixed-Integer Convex Model

This paper addresses the problem regarding the optimal placement and sizing of distribution static synchronous compensators (D-STATCOMs) in electrical distribution networks via a stochastic mixed-integer convex (SMIC) model in the complex domain. The proposed model employs a convexification technique based on the relaxation of hyperbolic constraints, transforming the nonlinear mixed-integer programming model into a convex one. The stochastic nature of renewable energy and demand is taken into account in multiple scenarios with three different levels of generation and demand. The proposed SMIC model adds the power transfer losses of the D-STATOMs in order to size them adequately. Two objectives are contemplated in the model with the aim of minimizing the annual installation and operating costs, which makes it multi-objective. Three simulation cases demonstrate the effectiveness of the stochastic convex model compared to three solvers in the General Algebraic Modeling System. The results show that the proposed model achieves a global optimum, reducing the annual operating costs by 29.25, 60.89, and 52.54% for the modified IEEE 33-, 69-, and 85-bus test systems, respectively.

Mobility-Based Proactive Cache and Transmission Strategy in Millimeter Wave Heterogeneous Networks

Millimeter wave (mmWave) frequencies with rich bandwidth can greatly alleviate the surging traffic demand of wireless communication from mobile video streaming. However, the harsh propagation environment at mmWave bands and user mobility make it hard to provide a reliable service. Multibeam transmissions can be taken as an essential approach for maintaining a robust mmWave link. Meanwhile, the introduction of coded caching can further reduce the end-to-end latency and backhaul traffic. Considering the interaction between caching and transmission in mobility scenario, we develop a joint design for mmWave heterogeneous networks (HetNets), by combining multibeam transmission and maximum distance separable (MDS)-coded caching. First, a joint optimization problem of cache placement and beam pair selection is formulated, in order to maximize the backhaul traffic offload in mmWave HetNets. To solve the mixed-timescale mixed-integer nonlinear programming (MINLP) problem, we then decompose the joint problem into the long-term mobility-based proactive content caching (CC) and short-term content delivery (CD) subproblems. Subsequently, two many-to-many matching-based algorithms are proposed for tackling the CD subproblem and the convex technique is applied to deal with the CC subproblem. Simulation results demonstrate the efficacy of the proposed algorithms over benchmark schemes.

Cooperative optimal guidance of hypersonic glide vehicles by real-time optimization and deep learning

This paper investigates the cooperative optimal guidance of hypersonic glide vehicles (HGVs) in the terminal phase with consideration on time-varying velocity, aerodynamic forces, and practical constraints. The cooperative optimal guidance problem is formulated as an optimal control problem with time as the independent variable, which brings great convenience in controlling the impact time. We propose proper convexification techniques to convexify this problem and apply successive convex optimization to get the solution of the original problem. To achieve cooperative guidance of multiple HGVs with time coordination constraint, a deep learning–based approach is proposed to find an optimal common impact time assigned to all the HGVs. The training samples required by deep learning are obtained by convex optimization. An algorithm is then presented to summarize the cooperative optimal guidance strategy. In each guidance loop, the common impact time is updated according to mission conditions, and the guidance commands are generated by the successive solution procedure in real time. Numerical examples will be provided to demonstrate that the proposed cooperative optimal guidance algorithm is effective and efficient, and it can achieve better performance than a popular cooperative proportional navigation guidance law.

On the strength of recursive McCormick relaxations for binary polynomial optimization

Recursive McCormick relaxations are among the most popular convexification techniques for binary polynomial optimization. It is well-understood that both the quality and the size of these relaxations depend on the recursive sequence and finding an optimal sequence amounts to solving a difficult combinatorial optimization problem. We prove that any recursive McCormick relaxation is implied by the extended flower relaxation, a linear programming relaxation, which for binary polynomial optimization problems with fixed degree can be solved in strongly polynomial time.

Nonlocal Pseudo-Parabolic Equation with Memory Term and Conical Singularity: Global Existence and Blowup

Considered herein is the initial-boundary value problem for a semilinear parabolic equation with a memory term and non-local source wt−ΔBw−ΔBwt+∫0tg(t−τ)ΔBw(τ)dτ=|w|p−1w−1|B|∫B|w|p−1wdx1x1dx′ on a manifold with conical singularity, where the Fuchsian type Laplace operator ΔB is an asymmetry elliptic operator with conical degeneration on the boundary x1=0. Firstly, we discuss the symmetrical structure of invariant sets with the help of potential well theory. Then, the problem can be decomposed into two symmetric cases: if w0∈W and Π(w0)&gt;0, the global existence for the weak solutions will be discussed by a series of energy estimates under some appropriate assumptions on the relaxation function, initial data and the symmetric structure of invariant sets. On the contrary, if w0∈V and Π(w0)&lt;0, the nonexistence of global solutions, i.e., the solutions blow up in finite time, is obtained by using the convexity technique.

An exact algorithm for the static pricing problem under discrete mixed logit demand

Price differentiation is a common strategy in many markets. In this paper, we study a static multiproduct price optimization problem with demand given by a discrete mixed multinomial logit model. By considering a mixed logit model that includes customer specific variables and parameters in the utility specification, our pricing problem reflects well the discrete choice models used in practice. To solve this pricing problem, we design an efficient iterative optimization algorithm that asymptotically converges to the optimal solution. To this end, a linear optimization (LO) problem is formulated, based on the trust-region approach, to find a “good” feasible solution and approximate the problem from below. A convex optimization problem is designed using a convexification technique to approximate the optimization problem from above. Then, using a branching method, we tighten the optimality gap. The effectiveness of our algorithm is illustrated on several cases, and compared against solvers and existing state-of-the-art methods in the literature.