Inequality Constraints Research Articles

A gradient-based optimization algorithm to solve optimal control problems with conformable fractional-order derivatives

In this paper, we solve numerically a class of fractional optimal control problems in the conformable sense. We first propose a nonlinear fractional optimal control problem subject to general equality and inequality constraints. Then, based on a novel numerical integration scheme, we present a discretization method for the governed fractional-order system as well as the cost and constraint functionals, which yields a discretized approximate problem. We further derive the gradients of the cost and constraint functionals in regard to the discretized controls. On the basis of this result, a numerical optimization approach to solve the approximate problem is developed. Finally, three non-trivial example problems are solved to illustrate the applicability and effectiveness of the developed approach.

Ray tracing through absorbing dielectric media in the Schwarzschild spacetime

Abstract General Relativity describes the trajectories of light-rays through curved spacetime near a massive object. In addition to gravitational lensing, we include an absorbing dielectric medium given by a complex refractive index known as the Drude model. When absorption is included the eikonal becomes complex, with the imaginary part related to the absorption along a ray between emission and observation points. We extend results from the literature to include dispersion in the index of refraction. The complex Hamiltonian splits into a real part that describes the equations of motion and a constraint equation that governs the momentum loss in the system. We work in coordinates which are fully real, with a real metric in physical spacetime. We assume the dust and plasma distributions of the Drude matter to coincide and vary as a power-law $1/r^h$. We find that transmission requires $h>1$, otherwise exponential absorption occurs along ray paths. We use ray-tracing through strongly absorbing matter near the surface of the compact star, as well as specializing to a point-lens in the weak-field limit with weakly absorbing matter to generate potentially observable light curves for distant observers. In the appropriate limits, our theory reproduces results from the literature.

Stochastic linear–quadratic control problems with affine constraints

This paper investigates the stochastic linear–quadratic control problems with affine constraints, in which both equality and inequality constraints are involved. With the help of the Pontryagin maximum principle and Lagrangian duality theory, both the dual problem and the state feedback form of the solution are obtained for the primal problem. Under the Slater condition, the strong duality is proved between the dual problem and the primal problem, and the KKT condition is also provided for solving the primal problem. Moreover, a new sufficient condition is given for the invertibility assumption, which ensures the uniqueness of the solutions to the dual problem. Finally, two numerical examples are provided to illustrate our main results.

An improving integration-enhanced ZNN for solving time-varying polytope distance problems with inequality constraint

An improving integration-enhanced ZNN for solving time-varying polytope distance problems with inequality constraint

Uniform stretching behavior of single crease origami arrays

Origami arrays featuring ideal purely rotational creases inherently possess multiple kinematic degrees of freedom. The incorporation of crease rotational stiffness introduces additional constraint relations to the kinematic morphology of the origami array, due to the interaction between the plate and the crease. This paper concentrates on elucidating the coupling effect, employing two theoretical models to systematically analyze the uniform stretch process of single crease origami arrays. The validity of our theoretical approaches is substantiated through verification in the finite element method. Firstly, the mechanical equations of the origami unit, together with the length and angle constraint equations of the origami array, are amalgamated to formulate a comprehensive set of morphological computational equations specific to the single crease origami array. This set enables the realization of morphological analysis for general origami arrays. Moreover, energy equations for the origami array are computed using an equivalent rigid plate nonlinear creasing element. When combined with constraint equations corresponding to horizontal stretch displacement, these energy equations contribute to the establishment of a nonlinear optimization framework for determining the morphology of uniform single crease origami arrays. Notably, our investigation reveals that during the early stages of the unfolding process for a single crease origami array, even a small relative horizontal stretch displacement can induce a drastic change in the center crease. Through finite element analysis of general origami arrays, it is shown that the proposed theoretical analysis method applies to different crease stiffness, both unfolding and folding processes, and different plate types. The research method in this paper can provide ideas for systematically analyzing the influence mechanism of crease properties on the morphological features of origami structures.

A Moment-Sum-of-Squares Hierarchy for Robust Polynomial Matrix Inequality Optimization with Sum-of-Squares Convexity

We study a class of polynomial optimization problems with a robust polynomial matrix inequality (PMI) constraint where the uncertainty set itself is also defined by a PMI. These can be viewed as matrix generalizations of semi-infinite polynomial programs because they involve actually infinitely many PMI constraints in general. Under certain sum-of-squares (SOS)-convexity assumptions, we construct a hierarchy of increasingly tight moment-SOS relaxations for solving such problems. Most of the nice features of the moment-SOS hierarchy for the usual polynomial optimization are extended to this more complicated setting. In particular, asymptotic convergence of the hierarchy is guaranteed, and finite convergence can be certified if some flat extension condition holds true. To extract global minimizers, we provide a linear algebra procedure for recovering a finitely atomic matrix-valued measure from truncated matrix-valued moments. As an application, we are able to solve the problem of minimizing the smallest eigenvalue of a polynomial matrix subject to a PMI constraint. If SOS convexity is replaced by convexity, we can still approximate the optimal value as closely as desired by solving a sequence of semidefinite programs and certify global optimality in case that certain flat extension conditions hold true. Finally, an extension to the nonconvexity setting is provided under a rank 1 condition. To obtain the above-mentioned results, techniques from real algebraic geometry, matrix-valued measure theory, and convex optimization are employed. Funding: This work was supported by the National Key Research and Development Program of China [Grant 2023YFA1009401] and the National Natural Science Foundation of China [Grants 11571350 and 12201618].

Equality‐embedded augmented Lagrangian neural network for DC optimal power flow

AbstractDirect current optimal power flow (DC‐OPF) problems need to be solved more frequently to maintain safety and economic power system operation. Traditional solvers take too much time to get optimal results. To overcome it, a new self‐supervised augmented Lagrangian neural network (ALNN) is proposed to solve DC‐OPF problem. The proposed ALNN consists of two neural networks: the control net and the penalty net. The control net predicts active power of generators; the penalty net updates the Lagrangian multipliers. The equality constraints are embedded into the control net to guarantee no equality violations. The generalized reduced gradient method is used to reduce theviolations of inequality constraint. The effectiveness of the proposed model is demonstrated on IEEE 118‐bus. The results show that with the help of equality embedding, the equality constraints are always satisfied, which in turn improves the feasibility of ALNN. Compared to the state‐of‐art models, the proposed model has higher feasibility and less constraint violations without comprising optimality. What is more, most of the inactive constraints can be found during the training process and then they are used to speed up the post‐processing part.

Extended Lambert-Based Approach for Ground-Track Adjustment with Specified Overflight Altitude

This paper presents an extended Lambert-based approach for ground-track adjustment under J2 perturbation, aiming to achieve a satellite flyby over a designated ground location at a predetermined altitude. Extended equations are developed in the orbital plane based on Lagrange’s form for the unperturbed Lambert’s problem to address the J2-perturbed two-body motion. By combining them with the out-of-plane flight time equation constraint regarding the ground-track adjustment, a nonlinear system of equations with two unknowns and two equations is established and split into different modes. To solve these modes, a modified Newton–Raphson method is used, and a necessary condition is established to determine the maximum allowable number of revolutions. To enhance computational efficiency, a radius restriction method is proposed to narrow down the allowable range of the number of revolutions. The proposed algorithm is applicable to various scenarios, including single- or multiple-day flights, ascending or descending overflights, coplanar or non-coplanar transfers, and initial or reverse-initial directional transfers. The simulation results demonstrate that the proposed algorithm is effective and accurate for the perturbed adjustment problem.

Hierarchical Robust Generalized Nash Equilibrium Seeking of High-Order Uncertain Nonlinear Systems.

This article investigates the distributed generalized Nash equilibrium (GNE) seeking problem of noncooperative games (NGs) for high-order strict-feedback nonlinear multiagent systems (MASs). In particular, the feasible action set of each agent is not only subject to local set and inequality constraints but also coupled through an equality constraint with other agents. This constraint structure is more general and covers most of the constraints in the GNE seeking literature. To accomplish the concerned GNE seeking objective, we propose a novel hierarchical GNE seeking approach in this article to decouple the distributed GNE seeking algorithm design into two layers. First, we construct a distributed primary-dual GNE estimator to generate virtual reference signals that converge to the GNE. Then, with the output of the estimator as the reference signal, we develop an adaptive tracking controller to solve the resultant tracking problems under output constraints. To overcome the negative effects of the disturbances, novel compensating terms associated with smooth functions and positive integrable time-varying functions are incorporated in the controller design, which thereby realizes the exact GNE seeking in the presence of nonvanishing mismatched disturbances. At last, an example is given to support the theoretical analysis of the proposed algorithms.

Distributed Frank-Wolfe Solver for Stochastic Optimization With Coupled Inequality Constraints.

Distributed stochastic optimization (DSO) with local set constraints and coupled inequality constraints over a multiagent network is considered in this article. Usually, such problems are tackled by projected primal-dual methods, which require expensive projection operations when set constraints are complicated. In this context, this article focuses on the Frank-Wolfe (FW) framework, which provides computational simplicity by avoiding expensive projection operations, for solving DSO with local set and coupled inequality constraints. By combining recursive momentum and weighted averaging, this article proposes a distributed stochastic FW primal-dual algorithm (DSFWPD), which is the first stochastic FW solver for DSO problems with coupled constraints. The proposed algorithm achieves zero constraint violation on average with a sublinear decay of the optimality gap over a directed and time-varying network. The efficacy of DSFWPD is demonstrated by several numerical experiments.

Two Acceleration-Layer Configuration Amendment Schemes of Redundant Robot Arms Based on Zhang Neurodynamics Equivalency.

Two innovative acceleration-layer configuration amendment (CA) schemes are proposed to achieve the CA of constrained redundant robot arms. Specifically, by applying the Zhang neurodynamics equivalency (ZNE) method, an acceleration-layer CA performance indicator is derived theoretically. To obtain a unified-layer inequality constraint by transforming from angle-layer and velocity-layer constraints to acceleration-layer constraints, five theorems and three corollaries are theoretically derived and rigorously proved. Then, together with the unified acceleration-layer bound constraint, an enhanced acceleration-layer CA scheme specially considering three-layer time-variant physical limits is proposed, and a simplified acceleration-layer CA scheme considering three-layer time-invariant physical limits is also proposed. The proposed CA schemes are finally formulated in the form of standard quadratic programming and are solved by a projection neurodynamics solver. Moreover, comparative simulative experiments based on a four-link planar arm and a UR3 spatial arm are performed to verify the efficacy and superiority of the proposed CA schemes. At last, physical experiments are conducted on a real Kinova Jaco2 arm to substantiate the practicability of the proposed CA schemes.

A statistic-based tracking method of equivalent constraint position for monopile supported offshore wind turbine

Bearing capacity of pile foundation is one of the important factors affecting the safe operation of the offshore wind turbines. However, due to the influence of different geological conditions, pile types and other factors, it is difficult for the existing monitoring methods to obtain the change of real constraints of the pile foundation. A statistic-based equivalent constraint position prediction method for offshore wind turbines is proposed. The method is based on the first-order displacement and first-order frequency extracting from the measured acceleration signals obtained by the accelerometers installed above the water surface, and statistical method is adopted to eliminate the influence of noise and reduce the error caused by single prediction. The most achievement of the proposed method is that the constraint of pile foundation related to bearing capacity is obtained and quantified by equivalent constraint position, and more accurate assessment of safety state of the offshore wind turbine can be achieved. To verify the reliability of proposed method, numerical model of a 4-MW monopile supported offshore wind turbine considering the pile–soil interaction is carried out. The results show that the proposed method can accurately obtain the equivalent constraint position by a combination of first-order frequency and first-order displacement. Further, the field measurement of an offshore wind turbine located in Rudong County, Jiangsu Province of China is performed. A stable prediction of equivalent constraint range from -30.86 m to -30.15 m is obtained over 43 days of the test. Therefore, the proposed method can be used to predict the change of pile foundation constraint, and is of great significance to the safety assessment of offshore wind turbines.

A novel improved harbor seal whiskers algorithm for solving hybrid dynamic economic environmental dispatch considering uncertainty of renewable energy generation

The originality of this work is introducing a novel Improved Harbor Seal Whiskers Algorithm (IHSWA) as an innovative and improved optimizer for solving complex hybrid dynamic economic-environmental dispatch (HDEED) considering uncertainty of renewable energy generation (REG) and various system constraints. IHSWA is implemented by hybridizing opposition-based algorithm (OBA) and local search algorithm (LSA) approaches to enhance the algorithm's search efficiency and overcome the challenges of other counterpart approaches. IHSWA has an optimal balancing feature between searching modes (exploring and exploiting modes). This feature makes the proposed algorithms more advanced than their counterpart optimizing techniques. The HDEED is modeled using a non-convex and non-smooth objective function, considering various equality and inequality constraints such as ramp rate bounds (RRLs), valve-point effects (VPE), production limits (PL), spinning reserves (SRLs), prohibited operating zones (POZs), and power balance (PB). The proposed approach is verified on a hybrid modified IEEE 57-bus 7-unit power system incorporating solar and wind energy generation. The proposed approach is compared with various modern optimization techniques, such as the traditional Harbor Seal Whiskers Algorithm (HSWA), the Lemurs Optimization Algorithm (LOA), the Nutcracker Optimization Algorithm (NOA), and the Artificial Hummingbird Algorithm (AHA). The incorporation of REG sources produces a significant reduction in operative costs and a remarkable decline in emissions. The convergence rate of the proposed approach indicates that the novel hybrid algorithm converges rapidly towards the optimal solution, achieving the best global optimum solution. The ANOVA test and the Tukey test are used to analyze the results statistically. The IHSWA approach shows superior advanced features with a best fuel cost of 1640.5890 $ and a minimum std. value of 1708.4895. While for emission objective minimizing, IHSWA attained a minimum emission cost and std. value of 1080.1629 kg and 1777.0770, respectively. The statistical analysis of the results approves the stability, reliability, and consistency of the proposed strategy with significant convergence. The appreciated contribution of this work is introducing a novel hybrid approach for solving HDEED efficiently, incorporating REG, and considering various system constraints. IHSWA has advanced features over its counterpart approaches. The contribution holds specific importance for the economic development of countries. It creates the path for increased self-sufficiency by reducing their need for fossil fuels for energy production. The study emphasizes the transition from being reliant on fossil fuels to the development of sustainable urban economies.

Examples of cosmological spacetimes without CMC Cauchy surfaces

CMC (constant mean curvature) Cauchy surfaces play an important role in mathematical relativity as finding solutions to the vacuum Einstein constraint equations is made much simpler by assuming CMC initial data. However, Bartnik (Commun Math Phys 117(4):615–624, 1988) constructed a cosmological spacetime without a CMC Cauchy surface whose spatial topology is the connected sum of two three-dimensional tori. Similarly, Chruściel et al. (Commun Math Phys 257(1):29–42, 2005) constructed a vacuum cosmological spacetime without CMC Cauchy surfaces whose spatial topology is also the connected sum of two tori. In this article, we enlarge the known number of spatial topologies for cosmological spacetimes without CMC Cauchy surfaces by generalizing Bartnik’s construction. Specifically, we show that there are cosmological spacetimes without CMC Cauchy surfaces whose spatial topologies are the connected sum of any compact Euclidean or hyperbolic three-manifold with any another compact Euclidean or hyperbolic three-manifold. Analogous examples in higher spacetime dimensions are also possible. We work with the Tolman–Bondi class of metrics and prove gluing results for variable marginal conditions, which allows for smooth gluing of Schwarzschild to FLRW models.

Physics-informed neural networks with hard linear equality constraints

Surrogate modeling is used to replace computationally expensive simulations. Neural networks have been widely applied as surrogate models that enable efficient evaluations over complex physical systems. Despite this, neural networks are data-driven models and devoid of any physics. The incorporation of physics into neural networks can improve generalization and data efficiency. The physics-informed neural network (PINN) is an approach to leverage known physical constraints present in the data, but it cannot strictly satisfy them in the predictions. This work proposes a novel physics-informed neural network, KKT-hPINN, which rigorously guarantees hard linear equality constraints through projection layers derived from KKT conditions. Numerical experiments on Aspen models of a continuous stirred-tank reactor (CSTR) unit, an extractive distillation subsystem, and a chemical plant demonstrate that this model can further enhance the prediction accuracy.

Multi-segmented fifth-order polynomial–shaped shells under hydrostatic pressure

The design and construction of submerged complex shells for new applications in offshore structures are increasingly popular. Therefore, the purpose of this study is to present the analytical model and Lagrange multipliers associated with the constraint equation for large displacement analysis of a fifth-order polynomial–shaped shell under hydrostatic pressure for the first time. The shell geometry can be computed using differential geometry with a fifth-order polynomial. The energy functional of the fifth-order polynomial–shaped shell is derived based on the principle of virtual work and written in the appropriate form. The nonlinear static responses of the fifth-order polynomial–shaped shell under hydrostatic pressure can be calculated using the nonlinear finite element method via the fifth-order polynomial shape function. This study develops the model using one-dimensional beam elements divided along the shell radius. To avoid the slope of a meridian curve at the equatorial plane approaching infinity, the shell is divided into two regions defined by different surface parameters. At the junction of two adjacent regions, the continuity requirements are established as the constraint conditions using Lagrange multipliers. The numerical results from the proposed methods are demonstrated and discussed, along with the effects of varied seawater depth, thickness, and elastic modulus on the deformed configuration and principal curvature at the deformed state. The results show that the nonlinear displacement is higher than the linear one in the case of the hydrostatic pressure, whereas the case of the internal pressure has an opposite result. For principal curvatures at the apex, the principal curvatures increase as the seawater depth increases, whereas the principal curvatures decrease when the thickness and elastic modulus increase.

The linearized Einstein equations with sources

On vacuum spacetimes of general dimension, we study the linearized Einstein vacuum equations with a spatially compactly supported and (necessarily) divergence-free source. We prove that the vanishing of appropriate charges of the source, defined in terms of Killing vector fields on the spacetime, is necessary and sufficient for solvability within the class of spatially compactly supported metric perturbations. The proof combines classical results by Moncrief with the solvability theory of the linearized constraint equations with control on supports developed by Corvino–Schoen and Chruściel–Delay.

Multi-view hypergraph regularized Lp norm least squares twin support vector machines for semi-supervised learning

In recent years, multi-view semi-supervised learning has gradually become a popular research direction. The classic binary classification methods in this field are multi-view Laplacian support vector machines (MvLapSVM) and multi-view Laplacian twin support vector machines (MvLapTSVM), which extend semi-supervised support vector machine to multi-view learning. Nevertheless, similar to the majority of SVM-based multi-view methods, the above methods are two-view methods that cannot fully leverage the information from all views and are constructed based on the L2 norm. Additionally, in semi-supervised graph learning, the quality of the graph often has a significant impact on the results. Therefore, we propose a novel multi-view hypergraph regularized Lp norm least squares twin support vector machines (MvHGLpLSTSVM) that can handle general multi-view data for semi-supervised learning. It extends hypergraph learning to multi-view learning and combines Lp norm to further explore the manifold structure and embedded geometric information of multi-view data. By using equality constraints, we design a simple and effective iterative algorithm. In the classification of six multi-view datasets, we compare the proposed method with some other state-of-the-art methods, and the results show that the proposed method is effective.

Optimality and KKT conditions for interval valued optimization problems on Hadamard manifolds

Recently, a new type of optimization problems, the so-called interval optimization problems on Hadamard manifolds, is introduced by the authors in Nguyen et al. [Interval optimization problems on Hadamard manifolds. J Nonlinear Convex Anal. 2023;24(11):2489–2511]. In this follow-up, we further offer the algorithmic bricks for these problems. More specifically, we characterize the optimality and KKT conditions for the interval valued optimization problems on Hadamard manifolds. For unconstrained problems, the existence of efficient points and the steepest descent algorithm are investigated. To the contrast, the KKT conditions and exact penalty approach are explored in the ones involving inequality constraints. These results pave the foundations for the solvability of interval valued optimization problems on Hadamard manifolds.

Bound‐based control design for permanent magnet linear motor servo system with inequality constraints and uncertainties

AbstractThe permanent magnet linear motor (PMLM) systems are widely applied in various engineering fields. However, the state inequality constraints and uncertainties of parameters and unknown external disturbances need to be further explored. In this article, first, a state transformation method is proposed for converting bounded states to unbounded states. As a result, the output state of system is limited within the expected boundary constraint. Then, a new control strategy is constructed. This strategy can ensure that the controlled system with uncertainties is uniformly bounded and uniformly ultimately bounded. Finally, the effectiveness of the controller and the state transformation method are verified through simulation of PMLM system. The inequality constraints of the output state are satisfied, and the system performance is improved.