Second-order Sufficient Conditions Research Articles

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.

Second-order sufficient conditions in optimal control of evolution systems

Second-order sufficient conditions in optimal control of evolution systems

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.

To Charge In-Flight or Not: A Comparison of Two Parallel Hybrid Electric Aircraft Configurations via Optimal Control

This study examines two configurations for parallel hybrid electric aircraft, one with a mechanical connection between the engines and the electric motors and the other without. A previously proposed power allocation algorithm in the context of aircraft energy management is applied to these two configurations for a 19-seat conceptual hybrid electric aircraft. The original optimal control problem is then transformed into a finite-dimensional optimization, and the second-order sufficient conditions for the global optimality of the obtained solution are validated. This is followed by a sensitivity analysis of the fuel consumption to the initial aircraft weight and flight duration. Simulation and theoretical results clarify the limited benefits of charging the battery in-flight for this class of hybrid electric aircraft for the purpose of reducing fuel consumption and CO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> emissions.

Second-Order Sufficient Conditions in the Sparse Optimal Control of a Phase Field Tumor Growth Model with Logarithmic Potential

This paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn-Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term like the L1-norm in order to enhance the occurrence of sparsity effects in the optimal controls, i.e., of subdomains of the space-time cylinder where the controls vanish. In the context of cancer therapies, sparsity is very important in order that the patient is not exposed to unnecessary intensive medical treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for the optimal control problem. While in previous works on the system under investigation such conditions have been established for the case without sparsity, the case with sparsity has not been treated before.

Second-order sufficient conditions, Part 2: Weak local minima in an optimal control problem with a general control constraint

Second-order sufficient conditions, Part 2: Weak local minima in an optimal control problem with a general control constraint

Non-existence of perturbed solutions under a second-order sufficient condition

We present an optimization problem in infinite dimensions which satisfies the usual second-order sufficient condition but for which perturbed problems fail to possess solutions.

A Smoothing Method for Sparse Programs by Symmetric Cone Constrained Generalized Equations

In this paper, we consider a sparse program with symmetric cone constrained parameterized generalized equations (SPSCC). Such a problem is a symmetric cone analogue with vector optimization, and we aim to provide a smoothing framework for dealing with SPSCC that includes classical complementarity problems with the nonnegative cone, the semidefinite cone and the second-order cone. An effective approximation is given and we focus on solving the perturbation problem. The necessary optimality conditions, which are reformulated as a system of nonsmooth equations, and the second-order sufficient conditions are proposed. Under mild conditions, a smoothing Newton approach is used to solve these nonsmooth equations. Under second-order sufficient conditions, strong BD-regularity at a solution point can be satisfied. An inverse linear program is provided and discussed as an illustrative example, which verified the efficiency of the proposed algorithm.

Second-Order Optimality Conditions and Duality for Multiobjective Semi-Infinite Programming Problems on Hadamard Manifolds

This paper is devoted to the study of multiobjective semi-infinite programming problems on Hadamard manifolds. We consider a class of multiobjective semi-infinite programming problems (abbreviated as MSIP) on Hadamard manifolds. We use the concepts of second-order Karush–Kuhn–Tucker stationary point and second-order Karush–Kuhn–Tucker geodesic pseudoconvexity of the considered problem to derive necessary and sufficient second-order conditions of efficiency, weak efficiency and proper efficiency for MSIP along with certain generalized geodesic convexity assumptions. Moreover, we formulate the second-order Mond–Weir-type dual problem related to MSIP and deduce weak and strong duality theorems relating MSIP and the dual problem. The significance of our results is demonstrated with the help of non-trivial examples. To the best of our knowledge, this is the first time that second-order optimality conditions for MSIP have been studied in Hadamard manifold setting.

A simple proof of second-order sufficient optimality conditions in nonlinear semidefinite optimization

In this note, we present an elementary proof for a well-known second-order sufficient optimality condition in nonlinear semidefinite optimization which does not rely on the enhanced theory of second-order tangents. Our approach builds on an explicit elementary computation of the so-called second subderivative of the indicator function associated with the semidefinite cone which recovers the best curvature term known in the literature.

Correction: Second-Order Sufficient Optimality Conditions for an Optimal Control Problem with Mixed Constraints

Correction: Second-Order Sufficient Optimality Conditions for an Optimal Control Problem with Mixed Constraints

On the Strong Subregularity of the Optimality Mapping in an Optimal Control Problem with Pointwise Inequality Control Constraints

This paper presents sufficient conditions for strong metric subregularity (SMsR) of the optimality mapping associated with the local Pontryagin maximum principle for Mayer-type optimal control problems with pointwise control constraints given by a finite number of inequalities G_j(u)le 0. It is assumed that all data are twice smooth, and that at each feasible point the gradients G_j'(u) of the active constraints are linearly independent. The main result is that the second-order sufficient optimality condition for a weak local minimum is also sufficient for a version of the SMSR property, which involves two norms in the control space in order to deal with the so-called two-norm-discrepancy.

Local Optimality Conditions for a Family of Hidden Convex Optimization

Hidden convex optimization is a class of nonconvex optimization problems that can be globally solved in polynomial time via equivalent convex programming reformulations. In this paper, we study a family of hidden convex optimization that joints the classical trust region subproblem (TRS) with convex optimization (CO). It also includes p-regularized subproblem (p &gt; 2) as a special case. We present a comprehensive study on local optimality conditions. In particular, a sufficient condition is given to ensure that there is at most one local nonglobal minimizer, and at this point, the standard second-order sufficient optimality condition is necessary. To our surprise, although (TRS) has at most one local nonglobal minimizer and (CO) has no local nonglobal minimizer, their joint problem could have any finite number of local nonglobal minimizers. Funding: This work was supported by the National Natural Science Foundation of China [Grants 12171021, 12131004, and 11822103], the Beijing Natural Science Foundation [Grant Z180005], and the Fundamental Research Funds for the Central Universities. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2023.0089 .

Second-order optimality conditions for locally Lipschitz vector optimization problems

In this paper, we derive primal and dual second-order necessary and sufficient optimality conditions for a vector optimization problem with equality and inequality constraints where the functions involved are locally Lipschitz. We introduce a weaker notion of second-order Abadie constraint qualification to derive second-order necessary conditions for weak local Pareto minima and strict local Pareto minima of order two in terms of Páles and Zeidan's second-order upper directional derivatives. Dual necessary conditions are derived for both types of minimal solutions in finite-dimensional spaces assuming the functions to be first-order continuously Fréchet differentiable. In the same setting, we derive dual and primal second-order sufficient optimality conditions for strict local Pareto minima of order two in terms of second-order lower directional derivatives.

Second-order sufficient conditions for sparse optimal control of singular Allen–Cahn systems with dynamic boundary conditions

Second-order sufficient conditions for sparse optimal control of singular Allen–Cahn systems with dynamic boundary conditions

On the Convergence of Overlapping Schwarz Decomposition for Nonlinear Optimal Control

We study the convergence properties of an overlapping Schwarz decomposition algorithm for solving nonlinear optimal control problems (OCPs). The algorithm decomposes the time domain into a set of overlapping subdomains, and solves all subproblems defined over subdomains in parallel. The convergence is attained by updating primal-dual information at the boundaries of overlapping subdomains. We show that the algorithm exhibits local linear convergence, and that the convergence rate improves exponentially with the overlap size. We also establish global convergence results for a general quadratic programming, which enables the application of the Schwarz scheme inside second-order optimization algorithms (e.g., sequential quadratic programming). The theoretical foundation of our convergence analysis is a sensitivity result of nonlinear OCPs, which we call “exponential decay of sensitivity” (EDS). Intuitively, EDS states that the impact of perturbations at domain boundaries (i.e., initial and terminal time) on the solution decays exponentially as one moves into the domain. Here, we expand a previous analysis available in the literature by showing that EDS holds for both primal and dual solutions of nonlinear OCPs, under uniform second-order sufficient condition, controllability condition, and boundedness condition. We conduct experiments with a quadrotor motion planning problem and a partial differential equations (PDE) control problem to validate our theory, and show that the approach is significantly more efficient than alternating direction method of multipliers and as efficient as the centralized interior-point solver.

Error Estimates for Approximate Solutions to Seminlinear Elliptic Optimal Control Problems with Nonlinear and Mixed Constraints

This paper gives some sufficient conditions for convergence of approximate solutions to seminlinear elliptic optimal control problems with mixed pointwise constraints. We build discrete optimal control problems by the finite element method in type of the full control discretization. We show that if the strictly second-order sufficient condition is valid, then some error estimates between approximate solutions of discrete optimal control problems and optimal solutions of the original problem are obtained.

Second-Order Sufficient Optimality Conditions for an Optimal Control Problem with Mixed Constraints

Second-Order Sufficient Optimality Conditions for an Optimal Control Problem with Mixed Constraints

A second-order sufficient condition for a weak local minimum in an optimal control problem with an inequality control constraint

Abstract This paper is devoted to a sufficient second-order condition for a weak local minimum in a simple optimal control problem with one control constraint G(u) ≤ 0, given by a C 2-function. A similar second-order condition was obtained earlier by the author for a strong minimum in a much more general problem. In the present paper, we would like to take a narrower perspective than before and thus provide shorter and simpler proofs. In addition, the paper uses the first and second order tangents to the set U, defined by the inequality G(u) ≤ 0. The main difficulty of the proof, clearly shown in the paper, refers to the set, where the gradient H u of the Hamiltonian is small, but the condition of quadratic growth of the Hamiltonian is satisfied. The paper can be valuable for self-explanation and provides a basis for extensions.

Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs

We study solution sensitivity for nonlinear programs (NLPs) whose structures are induced by graphs. These NLPs arise in many applications such as dynamic optimization, stochastic optimization, optimization with partial differential equations, and network optimization. We show that for a given pair of nodes, the sensitivity of the primal-dual solution at one node against a data perturbation at the other node decays exponentially with respect to the distance between these two nodes on the graph. In other words, the solution sensitivity decays as one moves away from the perturbation point. This result, which we call exponential decay of sensitivity, holds under the strong second-order sufficiency condition and the linear independence constraint qualification. We also present conditions under which the decay rate remains uniformly bounded; this allows us to characterize the sensitivity behavior of NLPs defined over subgraphs of infinite graphs. The theoretical developments are illustrated with numerical examples.