Order Sufficient Optimality Conditions Research Articles

New second order sufficient optimality conditions for state constrained parabolic control problems

We study a control problem governed by a semilinear parabolic equation with pointwise control and state constraints imposed at every point of the space-time cylinder. We obtain second order sufficient optimality conditions for local optimality in the sense of L 2 ( Q ) . Our results are valid for spatial domains of dimension less than or equal to three.

On the Second Order Sufficient Optimality Conditions for a Problem of Mathematical Programming

On the Second Order Sufficient Optimality Conditions for a Problem of Mathematical Programming

A Priori Error Analysis for an Optimal Control Problem Governed by a Variational Inequality of the Second Kind

We consider an optimal control problem governed by an elliptic variational inequality of the second kind. The problem is discretized by linear finite elements for the state and a variational discrete approach for the control. Based on a quadratic growth condition we derive nearly optimal a priori error estimates. Moreover, we establish second order sufficient optimality conditions that ensure a quadratic growth condition. These conditions are rather restrictive, but allow us to construct a one-dimensional locally optimal solution with reduced regularity, which serves as an exact solution for numerical experiments.

State Error Estimates for the Numerical Approximation of Sparse Distributed Control Problems in the Absence of Tikhonov Regularization

In this paper, we analyze optimal control problems of semilinear elliptic equations, where the controls are distributed. Box constraints for the controls are imposed and the cost functional does not involve the control itself, except possibly for a non-differentiable sparsity-promoting term. Under appropriate second order sufficient optimality conditions, first we estimate the difference between the discrete and continuous optimal states. Next, under an additional assumption on the optimal adjoint state, we prove error estimates for the controls and improve the estimates for the states.

On the strong subregularity of the optimality mapping in mathematical programming and calculus of variations

The paper presents sufficient conditions for a strong metric subregularity (SMSr) property of the optimality mapping associated with the Pontryagin local maximum principle for a Mayer's type optimal control problem with general initial/terminal constraints for the state variable and unconstrained control. This SMSr property is adapted to the involvement of two norms in the basic assumptions: smoothness, constraint qualification, and strong second order sufficient optimality conditions. The proofs are based on a new abstract result for strong metric subregularity (in a two-norms setting) of the Karush-Kuhn-Tucker optimality mapping for a mathematical programming problem in a Banach space, also presented in the paper.

Penalization of Dirichlet Boundary Control for Nonstationary Magneto-Hydrodynamics

Penalization of Dirichlet boundary controlled nonstationary magneto-hydrodynamic equations is considered. Asymptotic behavior of solutions of a penalized control problem with respect to the penalty parameter is investigated. It is proved that solutions of the penalized boundary control problem converge to the solutions of the Dirichlet control problem as penalty parameter $\epsilon$ goes to zero. The existence of optimal solutions as well as the characterization of such solutions by first order necessary conditions are established. A second order sufficient optimality condition for the optimal control problem is also developed. Numerical results are provided showing the feasibility and effectiveness of the penalty method.

Critical Cones for Sufficient Second Order Conditions in PDE Constrained Optimization

In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for different kinds of minima: strong or weak minimizers for optimal control problems. After a discussion on these extensions, we propose a new extended cone smaller than those considered until now. We prove that a second order condition based on this new cone is sufficient for a strong local minimum.

On the characterizations of solutions to perturbed l1 conic optimization problem

ABSTRACTThis paper focuses on perturbation analysis of the conic optimization problem, which is defined as the optimization problem over the epigraph of the weighted norm. The motivation for studying such problem comes from recent interest in the regularized (possibly non-convex) optimization problems arising in a wide variety of fields such as compressive sensing, signal processing and statistical learning. This paper first derives some important geometrical properties of relevant closed convex cone, including the tangent cone, the normal cone and the critical cone. We then show that under the Robinson's constraint qualification, the following conditions are equivalent: the constraint nondegeneracy and the strong second order sufficient optimality condition, the strong regularity of the Karush–Kuhn–Tucker (KKT) point, and the nonsingularity of Clarke's generalized Jacobian of the KKT system, and others. We further provide an important characterization of the isolated calmness for the conic optimization problem, namely, under the Robinson's constraint qualification, the isolated calmness of the KKT solution mapping holds if and only if the strict constraint qualification and the second order sufficient condition hold at a locally optimal solution. These characterizations provide theoretical results to design and analyse efficient algorithms for the conic optimization problem.

A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Time-Optimal Control Problems

Space-time finite element discretizations of time-optimal control problems governed by linear parabolic PDEs and subject to pointwise control constraints are considered. Optimal a priori error estimates are obtained for the control variable based on a second order sufficient optimality condition.

A convergence analysis of the method of codifferential descent

This paper is devoted to a detailed convergence analysis of the method of codifferential descent (MCD) developed by professor V.F. Demyanov for solving a large class of nonsmooth nonconvex optimization problems. We propose a generalization of the MCD that is more suitable for applications than the original method, and that utilizes only a part of a codifferential on every iteration, which allows one to reduce the overall complexity of the method. With the use of some general results on uniformly codifferentiable functions obtained in this paper, we prove the global convergence of the generalized MCD in the infinite dimensional case. Also, we propose and analyse a quadratic regularization of the MCD, which is the first general method for minimizing a codifferentiable function over a convex set. Apart from convergence analysis, we also discuss the robustness of the MCD with respect to computational errors, possible step size rules, and a choice of parameters of the algorithm. In the end of the paper we estimate the rate of convergence of the MCD for a class of nonsmooth nonconvex functions that arises, in particular, in cluster analysis. We prove that under some general assumptions the method converges with linear rate, and it convergence quadratically, provided a certain first order sufficient optimality condition holds true.

Higher order optimality and duality in fractional vector optimization over cones

In this paper we give higher order sufficient optimality conditions for a fractional vector optimization problem over cones, using higher order cone-convex functions. A higher order Schaible type dual program is formulated over cones.Weak, strong and converse duality results are established by using the higher order cone convex and other related functions.

Generalized Higher Order (φ, η, ω, π, ρ, θ, m)-Invexities in Parametric Optimality Conditions for Discrete Minmax Fractional Programming

First several new classes of higher order (φ, η, ω, π, ρ, θ, m)-invexities are introduced, and then a set of higher-order parametric necessary optimality conditions and several sets of higher order sufficient optimality conditions for a discrete minmax fractional programming problem applying various higher order (φ, η, ω, π, ρ, θ, m)-invexity constraints are established. The obtained results are new and generalize a wide range of results in the literature.

Second order analysis for the optimal control of parabolic equations under control and final state constraints

We consider the optimal control of a semilinear parabolic equation with pointwise bound constraints on the control and finitely many integral constraints on the final state. Using the standard Robinson’s constraint qualification, we provide a second order necessary condition over a set of strictly critical directions. The main feature of this result is that the qualification condition needed for the second order analysis is the same as for classical finite-dimensional problems and does not imply the uniqueness of the Lagrange multiplier. We establish also a second order sufficient optimality condition which implies, for problems with a quadratic Hamiltonian, the equivalence between solutions satisfying the quadratic growth property in the L 1 and $L^{\infty }$ topologies.

Error estimates for optimal control problems of a class of quasilinear equations arising in variable viscosity fluid flow

We consider optimal control problems of quasilinear elliptic equations with gradient coefficients arising in variable viscosity fluid flow. The state equation is monotone and the controls are of distributed type. We prove that the control-to-state operator is twice Frechet differentiable for this class of equations. A finite element approximation is studied and an estimate of optimal order h is obtained for the control. The result makes use of the distributed structure of the controls, together with a regularity estimate for elliptic equations with Holder coefficients and a second order sufficient optimality condition. The paper ends with a numerical experiment, where the approximation order is computationally tested.

Second Order and Stability Analysis for Optimal Sparse Control of the FitzHugh--Nagumo Equation

Optimal sparse control problems are considered for the FitzHugh--Nagumo system including the so-called Schlogl model. The nondifferentiable objective functional of tracking type includes a quadratic Tikhonov regularization term and the $L^1$-norm of the control that accounts for the sparsity. Though the objective functional is not differentiable, a theory of second order sufficient optimality conditions is established for Tikhonov regularization parameter $\nu>0$ and also for the case $\nu = 0$. In this context, also local minima are discussed that are strong in the sense of the calculus of variations. The second order conditions are used as the main assumption for proving the stability of locally optimal solutions with respect to $\nu \to 0$ and with respect to perturbations of the desired state functions. The theory is confirmed by numerical examples that are resolved with high precision to confirm that the optimal solution obeys the system of necessary optimality conditions.

Optimal Control of a Free Boundary Problem with Surface Tension Effects: A Priori Error Analysis

We present a finite element method along with its analysis for the optimal control of a model free boundary problem with surface tension effects, formulated and studied in [H. Antil, R. H. Nochetto, and P. Sodré, SIAM J. Control Optim., 52 (2014), pp. 2771--2799]. The state system couples the Laplace equation in the bulk with the Young--Laplace equation on the free boundary to account for surface tension. We first prove that the state and adjoint systems have the requisite regularity for the error analysis (strong solutions). We discretize the state, adjoint, and control variables via piecewise linear finite elements and show optimal $O(h)$ error estimates for all variables, including the control. This entails using the second order sufficient optimality conditions of [H. Antil, R. H. Nochetto, and P. Sodré, SIAM J. Control Optim., 52 (2014), pp. 2771--2799] and the first order necessary optimality conditions for both the continuous and discrete systems. We conclude with two numerical examples which examine the various error estimates.

Second-order sufficient optimality conditions for optimal control of static elastoplasticity with hardening

The paper is concerned with the optimal control of static elastoplasticity with linear kinematic hardening. This leads to an optimal control problem governed by an elliptic variational inequality (VI) of first kind in mixed form. Based on L p -regularity results for the state equation, it is shown that the control-to-state operator is Bouligand differentiable. This enables to establish second- order sufficient optimality conditions by means of a Taylor expansion of a particularly chosen Lagrange function.

Second Order Optimality Conditions and Their Role in PDE Control

If \(f: \mathbb{R}^{n} \to \mathbb{R}\) is twice continuously differentiable, f′(u)=0 and f″(u) is positive definite, then u is a local minimizer of f. This paper surveys the extension of this well known second order sufficient optimality condition to the case \(f: U \to \mathbb{R}\), where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How have second order sufficient optimality conditions to be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled?

Weakly monotone solutions of the Hamilton-Jacobi inequality and optimality conditions with positional controls

We obtain necessary global optimality conditions for classical optimal control problems based on positional controls. These controls are constructed with classical dynamical programming but with respect to upper (weakly monotone) solutions of the Hamilton-Jacobi equation instead of a Bellman function. We put special emphasis on the positional minimum condition in Pontryagin formalism that significantly strengthens the Maximum Principle for a wide class of problems and can be naturally combined with first order sufficient optimality conditions with linear Krotov's function. We compare the positional minimum condition with the modified nonsmooth Kaśkosz-Lojasiewicz Maximum Principle. All results are illustrated with specific examples.

Second order sufficient optimality conditions for hybrid control problems with state jump

In this paper, an optimal control problem for a class of hybrid systems is considered. By introducing a new time variable and transforming the hybrid optimal control problem into an equivalent problem, second order sufficient optimality conditions for this hybrid problem are derived. It is shown that sufficient optimality conditions can be verified by checking the Legendre-Clebsch condition and solving some Riccati equations with certain boundary and jump conditions. An example is given to show the effectiveness of the main results.