Second-order Sufficient Optimality 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.

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.

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 Optimality Conditions for an Optimal Control Problem with Mixed Constraints

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

Time Optimal Control Problem of the 3D Navier-Stokes-α Equations

In this paper, we study an optimal control problem for the three-dimensional Navier-Stokes-α equations in bounded domains with Dirichlet boundary conditions, where the time needed to reach a desired state plays an essential role. We first prove the existence of optimal solutions. Then we establish the first-order and second-order necessary optimality conditions, and the second-order sufficient optimality conditions. The second-order optimality ones obtained in the paper seem to be optimal in the sense that the gap between them is minimal.

Second-order optimality conditions for an optimal control problem governed by a regularized phase-field fracture propagation model

We prove second-order optimality conditions for an optimal control problem of tracking-type for a time-discrete regularized phase-field fracture or damage propagation model. The energy minimization problem describing the fracture process contains a penalization term for violation of the irreversibility condition in the fracture growth process, as well as a viscous regularization corresponding to a time-step restriction in a temporal discretization of the problem. In the control problem, the energy minimization problem is replaced by its Euler–Lagrange equations. While the energy minimization functional is convex due to the viscous approximation, the associated Euler–Lagrange equations are of quasilinear type, making the control problem nonconvex. We prove second-order necessary as well as second-order sufficient optimality conditions without two-norm discrepancy.

Sufficiency and duality of set-valued fractional programming problems via second-order contingent epiderivative

In this paper, we establish second-order sufficient KKT optimality conditions of a set-valued fractional programming problem under second-order generalized cone convexity assumptions. We also prove duality results between the primal problem and second-order dual problems of parametric, Mond-Weir, Wolfe, and mixed types via the notion of second-order contingent epiderivative.

On the strong metric subregularity in mathematical programming

Abstract This note presents sufficient conditions for the property of strong metric subregularity (SMSr) of the system of first order optimality conditions for a mathematical programming problem in a Banach space (the Karush-Kuhn-Tucker conditions). The constraints of the problem consist of equations in a Banach space setting and a finite number of inequalities. The conditions, under which SMSr is proven, assume that the data are twice continuously Fréchet differentiable, the strict Mangasarian-Fromovitz constraint qualification is satisfied, and the second-order sufficient optimality condition holds. The obtained result extends the one known for finite-dimensional problems. Although the applicability of the result is limited to the Banach space setting (due to the twice Fréchet differentiability assumptions and the finite number of inequality constraints), the paper can be valuable due to the self-contained exposition, and provides a ground for extensions. One possible extension was recently implemented in Osmolovskii and Veliov (2021).

Second-order optimality conditions in locally Lipschitz multiobjective fractional programming problem with inequality constraints

Our purpose, in this article, is to establish second-order optimality conditions for a second-order strict local Pareto minimum and a weak local Pareto minimum for a locally Lipschitz multiobjective fractional programming problem with inequality constraints. We do not require here the data to be second-order differentiable in any case. We first provide some basic calculus rules for the two extended-real-valued mappings defined on an open set of a Banach space. We second establish primal and dual Karush–Kuhn–Tucker second-order necessary optimality conditions to such problem. We third present second-order sufficient optimality conditions which are very near to a dual Karush–Kuhn–Tucker second-order necessary optimality condition. Some examples demonstrate the applicability of the obtained main results.

Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity

We study an optimal control problem for the stationary Stokes equations with variable density and viscosity in a 2D bounded domain under mixed boundary conditions. On in-flow and out-flow parts of the boundary, nonhomogeneous Dirichlet boundary conditions are used, while on the solid walls of the flow domain, the impermeability condition and the Navier slip condition are provided. We control the system by the external forces (distributed control) as well as the velocity boundary control acting on a fixed part of the boundary. We prove the existence of weak solutions of the state equations, by firstly expressing the fluid density in terms of the stream function (Frolov formulation). Then, we analyze the control problem and prove the existence of global optimal solutions. Using a Lagrange multipliers theorem in Banach spaces, we derive an optimality system. We also establish a second-order sufficient optimality condition and show that the marginal function of this control system is lower semi-continuous.

On the no-gap second-order optimality conditions for a non-smooth semilinear elliptic optimal control

This work is concerned with second-order necessary and sufficient optimality conditions for optimal control of a non-smooth semilinear elliptic partial differential equation, where the nonlinearity is the non-smooth max-function and thus the associated control-to-state operator is, in general, not Gâteaux-differentiable. In addition to standing assumptions, two main hypotheses are imposed. The first one is the Gâteaux-differentiability at the considered control of the objective functional and it is precisely characterized by the vanishing of an adjoint state on the set of all zeros of the corresponding state. The second one is a structural assumption on the sets of all points at which the values of the interested state are ‘close’ to the non-differentiability point of the max-function. We then derive a ‘no-gap’ theory of second-order optimality conditions in terms of a second-order generalized derivative of the cost functional, i.e. for which the only change between necessary and sufficient second-order optimality conditions are between a strict and non-strict inequality.

Sufficient Optimality Conditions in Bilevel Programming

This paper is concerned with the derivation of first- and second-order sufficient optimality conditions for optimistic bilevel optimization problems involving smooth functions. First-order sufficient optimality conditions are obtained by estimating the tangent cone to the feasible set of the bilevel program in terms of initial problem data. This is done by exploiting several different reformulations of the hierarchical model as a single-level problem. To obtain second-order sufficient optimality conditions, we exploit the so-called value function reformulation of the bilevel optimization problem, which is then tackled with the aid of second-order directional derivatives. The resulting conditions can be stated in terms of initial problem data in several interesting situations comprising the settings where the lower level is linear or possesses strongly stable solutions.

Regularity of multipliers in second-order optimality conditions for semilinear elliptic control problems

A class of semilinear elliptic optimal control problems with mixed pointwise constraints is considered. Some criteria under which Lagrange multipliers are functions in -spaces are given. Based on the criteria, we establish first- and second-order necessary optimality conditions of KKT-type as well as second-order sufficient optimality conditions for the problem.

Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis

This paper concerns a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type including chemotaxis with possibly singular potentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.

Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth

In this paper, we study an optimal control problem for a nonlinear system of reaction–diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in H. Garcke, et al. [Math. Model. Methods Appl. Sci. 26 (2016) 1095–1148]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [P. Colli, et al. To appear in: Appl. Math. Optim. (2019)] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L1−norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [P. Colli, et al. To appear in: Appl. Math. Optim. (2019)], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions.

No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation

This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is finitely PC2 (continuous and C2 apart from finitely many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish “no-gap” second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided.