Second-order Conditions Research Articles

Moreau-Yosida regularization to optimal control of the monodomain model with pointwise control and state constraints in cardiac electrophysiology

In this work, we study the optimal control problem of a coupled reaction-diffusion system, which is a monodomain model in cardiac electrophysiology with pointwise bilateral control and state constraints. We adopt the Moreau-Yosida regularization as a penalization technique to deal with the state constraints. The regularized problem’s first-order optimality condition is derived. In addition, sufficient second-order optimality condition is derived for the regularized problem using the virtual control concept by proving equivalence between Moreau-Yosida regularization and the virtual control concept. The convergence of optimal controls of the regularized problems to the optimal control of the original problem is proved. Moreover, the semi-smooth Newton method for numerically finding the optimal solution to the regularization problem is presented. Finally, numerical experiments are conducted, and the results allow us to understand the extinction of the wave excitation in cardiac defibrillation in the presence of both control and state constraints.

A path planner enabling second-order continuity across multiple motion modes of the generalized bicycle model

In mobile robotics, the generalized bicycle model is notable for its agility and diverse motion capabilities. However, current path planning methods often overlook its potential for multiple motion modes, leading to impractical paths and collision risks in complex environments. To address this, our study introduces a novel path planning algorithm that utilizes the generalized bicycle model and ensures second-order continuity, crucial for smooth and safe navigation. We establish rigorous second-order continuity conditions for seamless motion mode transitions and derive criteria for each mode. Our algorithm integrates these conditions with the rapidly exploring random tree concept to generate optimal paths that accommodate the model’s mode-changing capabilities. Through tests in challenging environments, our method demonstrates improved performance, especially in narrow passages, enhancing path planning safety and efficiency for mobile robots. This advancement significantly contributes to the field of robotic path planning by offering a more dynamic and reliable approach to autonomous navigation.

A fast smoothing newton method for bilevel hyperparameter optimization for SVC with logistic loss

Support vector classification (SVC) with logistic loss has excellent theoretical properties in classification problems where the label values are not continuous. In this paper, we reformulate the hyperparameter selection for SVC with logistic loss as a bilevel optimization problem in which the upper-level problem and the lower-level problem are both based on logistic loss. The resulting bilevel optimization model is converted to a single-level nonlinear programming (NLP) based on the KKT conditions of the lower-level problem. Such NLP contains a set of nonlinear equality constraints and a simple lower-bound constraint. The second-order sufficient condition is characterized, which guarantees that the strict local optimizers are obtained. To solve such NLP, we apply the smoothing Newton method proposed in [Liang L, Sun D., Toh KC. A squared smoothing Newton method for semidefinite programming, 2023] to solve the KKT conditions, which contain one pair of complementarity constraints. We show that the smoothing Newton method has a superlinear convergence rate. Extensive numerical results verify the efficiency of the proposed approach and strict local minimizers can be achieved both numerically and theoretically. In particular, compared with other methods, our algorithm can achieve competitive results while consuming less time than other methods.

Second-order Karush/Kuhn–Tucker conditions and duality for constrained multiobjective optimization problems

In this paper, we investigate the second-order strong Karush/Kuhn–Tucker conditions and duality of a constrained multiobjective optimization problem (CMOP). Exploiting a second-order regularization condition, we obtain second-order strong KKT necessary conditions of Borwein-properly efficient solution of CMOP without convexity assumptions. Further, second-order sufficient conditions for the second-order KKT point to be an efficient solution of CMOP are derived under the generalized second-order convexity assumptions. Finally, we establish duality results between CMOP and its second-order Wolfe-type dual problem.

Second-Order Optimality Conditions for Interval-Valued Optimization Problem

In this paper, we discuss the second-order KKT-type optimality conditions for interval-valued optimization problem. To derive second-order necessary and sufficient conditions, we use two different methodologies: square slack variable method and the theory of geometric cone optimality conditions for interval-valued objective functions. Furthermore, we discuss two types of order relations LU and center–radius (C–R). To show the validity of our results, we have demonstrated some nontriv ial examples.

Analysis of energy dissipation in hyperbolic problems influenced by internal and boundary control mechanisms

ABSTRACT This article primarily focuses on the rational stabilisation of the wave equation, supplied with a second-order dynamical boundary condition of hyperbolic type, while considering an additional internal damping mechanism within the specified ring. To achieve rational decay rates of the associated energy, it is imperative to exponentially stabilise a portion of the domain using a global Ingham's-type estimate. This paper will subsequently illustrate how this partially localised exponential stabilisation, combined with a Bessel analysis, leads to a rational decrease in the overall energy of the system considered.

Parabolic Regularity of Spectral Functions

This paper is devoted to the study of the second-order variational analysis of spectral functions. It is well-known that spectral functions can be expressed as a composite function of symmetric functions and eigenvalue functions. We establish several second-order properties of spectral functions when their associated symmetric functions enjoy these properties. Our main attention is given to characterize parabolic regularity for this class of functions. It was observed recently that parabolic regularity can play a central rule in ensuring the validity of important second-order variational properties, such as twice epi-differentiability. We demonstrates that for convex spectral functions, their parabolic regularity amounts to that of their symmetric functions. As an important consequence, we calculate the second subderivative of convex spectral functions, which allows us to establish second-order optimality conditions for a class of matrix optimization problems. Funding: The research of A. Mohammadi is funded by a postdoctoral fellowship from Georgetown University. E. Sarabi is partially supported by the U.S. National Science Foundation [Grant DMS 2108546].

Axisymmetric stagnation-point flow of a fractional Oldroyd-B hybrid nanofluid over a vertical cylinder by improved Buongiorno's model

This paper investigated the unsteady axisymmetric stagnation-point flow, heat and mass transfer of Oldroyd-B hybrid nanofluid over a vertical cylinder by using Caputo time fractional derivatives for the first time, with the consideration of second-order velocity slip conditions. Improved Buongiorno's model was adopted to build the governing equations and finite difference combined with L1-algorithm was employed to solve the numerical solutions. Effects of key physical parameters on velocity, heat and mass transfer were presented graphically and discussed in detail. Outcomes show that the fractional derivatives are capable to describe the memory effect and thermoelasticity of viscoelastic nanofluids. Both Brownian motion and thermophoresis, especially Brownian motion, improve the heat transfer on the cylinder wall. In addition, the sensitivity of average Nusselt (or Sherwood) number to velocity fractional derivates α and β rises with increasing α and β, while the sensitivity to temperature fractional derivate γ declines with increasing γ. When γ increases from 0.2 to 0.3, the relative growth rate of average Nusselt number is about 9.26 %. This study provides a reference for exploring the stationary-point flow of fractional order nanofluids on vertical cylinders.

Second-order necessary and sufficient optimality conditions for multiobjective interval-valued nonlinear programming

In this article, the second-order optimality conditions for nonlinear programming with multiple interval-valued objective functions (in brief, NPMIOF) are studied. In the first part, the second- and first-order necessary optimality conditions for some types of efficient solutions of NPMIOF are established under of Abadie second- and first-order constraint qualifications. In the next part, we investigate both primal and dual second-order sufficient optimality conditions. Our second-order necessary conditions enhance the previous results. The second-order sufficient optimality conditions are new.

Beyond Reasonableness: Argumentative Virtues in Pragma-Dialectics

AbstractThe pragma-dialectical research program begins with a philosophical estate, in which a conception of reasonableness is offered that must serve as ground for the theoretical estate. Pragma-dialectics has produced many important insights in the theoretical estate, including the ideal model and the rules for critical discussions. However, here I will argue that the conception of reasonableness that the pragma-dialecticians adopt in the philosophical estate, based on anti-dogmatism, assumption of fallibilism and willingness to engage in critical discussion, is too narrow to support the whole system of pragma-dialectical rules. What the philosophical estate requires is a broad and rich conception of excellent performance in argumentative practice, which then the rules of the critical discussion are intended to capture systematically in the theoretical estate. In my view, a virtue approach to argumentation is the ideal framework for such a philosophical ground. Virtues such as intellectual empathy, intellectual honesty, faith in reason, or recognition of reliable authority, point towards aspects of a philosophical conception of excellent arguing that are absent in the pragma-dialectical view of reasonableness. Finally, I will argue that what pragma-dialecticians call “second-order conditions” for a critical discussion are better understood as minimal argumentative virtue, a basic degree of virtue that arguers are required to possess in order to be prepared to participate in a fruitful critical discussion. The possession of such a basic degree of argumentative virtue is, I believe, what we mean when we characterise someone as reasonable.

Second-order optimality conditions for set-valued optimization problems under the set criterion

Second-order optimality conditions for set-valued optimization problems under the set criterion

On the convergence of local solutions for the method quADAPT

ABSTRACT A classical solution approach to semi-infinite programming, which is easy to implement, is based on discretizing the semi-infinite index set. The Blankenship and Falk algorithm adaptively chooses a small discretization. On every iteration, a solution based on the current discretization is calculated. In a second step, the most violated constraint is determined and added to the discretization. In a previous work, the authors showed that the algorithm has a slow convergence and introduced a new method that exhibits a quadratic rate [Seidel and Küfer, An adaptive discretization method solving semi-infinite optimization problems with quadratic rate of convergence. Optimization. 2022;71(8):2211–2239]. In this paper, we further investigate the introduced method. The method implements new constraints that can cut off parts of the feasible set. We will study the effect on local minima. We assume that in each iteration local solutions to the discretized problems are computed. We will give an example showing that in general a limit point is not necessarily a local solution of the original semi-infinite problem, but only of an approximate problem. We then study second-order conditions and show that they coincide for both problems. We use this to develop conditions under which local solutions converge to a local solution in the limit. Finally, we present quadratic convergence results for the case of local solutions.

The Indirect Bang-Singular Algorithm (IBSA) for singular control problems with state-inequality constraints

In this study, we present the Indirect Bang-Singular Algorithm (IBSA), a straightforward computational framework developed to solve a wide range of Singular Optimal Control Problems (SOCP) with state-inequality constraints. The algorithm is a type of arc-classification technique which reformulates the SOCP as a nonlinear programming problem over the switching times, and possibly some other parameters such as the co-states’ initial values at the entry time to a singular interval. We derive the singular control feedback using the Pontryagin’s maximum principle and analyze the possibility of an interval where multiple controls are simultaneously singular. Furthermore, we incorporate the state-inequality constraints using the direct-adjoining method. Owing to the linear property of the co-state dynamics, the co-state variables and consequently, the singular controls are computed automatically using MATLAB’s symbolic platform. The nonlinear programming is constructed in a manner to circumvent the challenges posed by state-inequality constraints in more intricate scenarios involving singular controls expressed in terms of incomplete state-feedback functions. We also present several theorems that are integral to devising a straightforward computational approach for solving SOCPs. To assess the effectiveness of the proposed algorithm, we solve the following novel problems: (1) time–fuel-optimal commercial aircraft cruise flight in a vertical plane (i.e., with state-inequality constraint, a scalar singular control, and wind shear effects), and (2) the free-routing time–fuel-optimal commercial aircraft flight in a vertical plane (i.e., with state-inequality constraint, a dual-entry singular control, and wind shear effects). Notably, the optimality of the graphed results has been carefully inspected through first and second-order optimality conditions.

A fast second-order absorbing boundary condition for the linearized Benjamin-Bona-Mahony equation

A fast second-order absorbing boundary condition for the linearized Benjamin-Bona-Mahony equation

Augmented Lagrangian method for nonlinear circular conic programs: a local convergence analysis

In this paper, we analyse a local convergence of augmented Lagrangian method (ALM) for a class of nonlinear circular conic optimization problems. In light of the singular value decomposition, the Debreu theorem and the implicit function theorem, we prove that the sequence generated by ALM converges to a local minimizer in the linear convergence rate under the constraint nondegeneracy condition and the strong second-order sufficient condition, in which the ratio constant is proportional to 1 / τ , where τ is the associated penalty parameter with a given lower threshold. As a byproduct, we also derive explicit expressions of critical cone and its affine hull for the given nonlinear circular conic program.

A nonsmooth primal-dual method with interwoven PDE constraint solver

We introduce an efficient first-order primal-dual method for the solution of nonsmooth PDE-constrained optimization problems. We achieve this efficiency through not solving the PDE or its linearisation on each iteration of the optimization method. Instead, we run the method interwoven with a simple conventional linear system solver (Jacobi, Gauss–Seidel, conjugate gradients), always taking only one step of the linear system solver for each step of the optimization method. The control parameter is updated on each iteration as determined by the optimization method. We prove linear convergence under a second-order growth condition, and numerically demonstrate the performance on a variety of PDEs related to inverse problems involving boundary measurements.

First- and second-order optimality conditions for the control of infinite horizon Navier–Stokes equations

First and second-order optimality conditions for optimal control problems over the infinite time horizon subject to the Navier–Stokes equations are derived. The cost functional enhances temporal sparsity of the controls, which implies that the optimal controls shut down in finite time. The problem formulation also includes explicit constraints on the control which may be non-smooth and non-affine.

Local properties and augmented Lagrangians in fully nonconvex composite optimization

A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper investigates the versatile template of composite optimization without any convexity assumptions. First- and second-order optimality conditions are discussed. We highlight the difficulties that stem from the lack of convexity when dealing with necessary conditions in a Lagrangian framework and when considering error bounds. Building upon these characterizations, a local convergence analysis is delineated for a recently developed augmented Lagrangian method, deriving rates of convergence in the fully nonconvex setting.

Optimal distributed control for a Cahn-Hilliard-Darcy system with mass sources, unmatched viscosities and singular potential

We study a Cahn-Hilliard-Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the averaged velocity u as well as homogeneous Neumann boundary conditions for the phase function ϕ and the chemical potential µ. The source term in the convective Cahn-Hilliard equation contains a control R that can be thought as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two-dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with ϕ being strictly separated from the pure phases ±1. This well-posedness result enables us to characterize the control-to-state mapping S: R→ϕ. Then we obtain the existence of an optimal control, the Frechet differentiability of S and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.

Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions

In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green’s first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.