Second-order Optimality Conditions Research Articles

Radius Theorems for Monotone Mappings

For a Hilbert space X and a mapping $F: X\rightrightarrows X$ (potentially set-valued) that is maximal monotone locally around a pair $(\bar {x},\bar {y})$ in its graph, we obtain a radius theorem of the following kind: the infimum of the norm of a linear and bounded single-valued mapping B such that F + B is not locally monotone around $(\bar {x},\bar {y}+B\bar {x})$ equals the monotonicity modulus of F. Moreover, the infimum is not changed if taken with respect to B symmetric, negative semidefinite and of rank one, and also not changed if taken with respect to all functions f : X → X that are Lipschitz continuous around $\bar {x}$ and ∥B∥ is replaced by the Lipschitz modulus of f at $\bar {x}$ . As applications, a radius theorem is obtained for the strong second-order sufficient optimality condition of an optimization problem, which in turn yields a lower bound for the radius of quadratic convergence of the smooth and semismooth versions of the Newton method. Finally, a radius theorem is derived for mappings that are merely hypomonotone.

Necessary First- and Second-Order Optimality Conditions in Discrete Systems with a Delay in Control

In the paper, an optimal control problem with a delay in control is considered. Suggesting a new approach, Euler-type necessary optimality conditions and the linearized discrete maximum principle are established. Also, the second-order necessary optimality conditions (a) based on the second variation of the objective functional and (b) for quasi-singular controls are obtained. An example to illustrate the richness of content of the suggested approach is presented.

Strong Local Minimizers in Optimal Control Problems with State Constraints: Second-Order Necessary Conditions

This paper is devoted to second-order necessary optimality conditions for strong local minima for a Mayer type optimal control problem with a general control constraint $U \subset {\mathbb R}^m$, and state and final-point constraints described by a finite number of inequalities. We use the second-order linearization of a relaxed differential inclusion associated to the control system to find a convex subset of second-order tangents to the set of its trajectories. This leads to second-order necessary optimality conditions via a straightforward way, based on separation theorems.

Second-Order KKT Optimality Conditions for MultiObjective Optimal Control Problems

In this paper, we study second-order necessary and sufficient optimality conditions of Karush--Kuhn--Tucker-type for locally optimal solutions in the sense of Pareto to a class of multi-objective optimal control problems with mixed pointwise constraint. To deal with the problems, we first derive second-order optimality conditions for abstract multi-objective optimal control problems which satisfy the Robinson constraint qualification. We then apply the obtained results to our concrete problems. The proofs of obtained results are direct, self-contained without using scalarization techniques.

Second-order optimality conditions for cone-subarcwise connected set-valued optimization problems

The concept of a cone subarcwise connected set-valued map is introduced. Several examples are given to illustrate that the cone subarcwise connected set-valued map is a proper generalization of the cone arcwise connected set-valued map, as well as the arcwise connected set is a proper generalization of the convex set, respectively. Then, by virtue of the generalized second-order contingent epiderivative, second-order necessary optimality conditions are established for a point pair to be a local global proper efficient element of set-valued optimization problems. When objective function is cone subarcwise connected, a second-order sufficient optimality condition is also obtained for a point pair to be a global proper efficient element of set-valued optimization problems.

Hybrid optimal control problems for a class of semilinear parabolic equations

A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may change. First- and second-order optimality conditions are derived. The analysis is based on a reformulation involving a judiciously chosen transformation of the time domains. For autonomous systems and a time-independent integral cost, we prove that the Hamiltonian is constant in time when evaluated along the optimal controls and trajectories. A numerical example is provided.

Enforcing Model Adequacy in Real-Time Optimization via Dedicated Parameter Adaptation

Iterative real-time optimization schemes that employ modifier adaptation add bias and gradient correction terms to the model that is used for optimization. These affine corrections lead to meeting the first-order necessary conditions of optimality of the plant despite plant-model mismatch. However, since the added terms do not include curvature information, satisfaction of the second-order sufficient conditions of optimality is not guaranteed, and the model might be deemed inadequate for optimization. In the context of modifier adaptation, this paper proposes to include a dedicated parameter-estimation step such that also the second-order optimality conditions are met at the plant optimum. In addition, we propose a procedure to select the best parameters to adapt based on a local sensitivity analysis. A simulation study dealing with product maximization in a fed-batch reactor demonstrates that the proposed scheme can both select the right parameters and determine their values such that modifier adaptation can drive the plant to optimality fast and without oscillations.

Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations

In this article, we consider the Tikhonov regularization of an optimal control problem of semilinear partial differential equations with box constraints on the control. We derive a-priori regularization error estimates for the control under suitable conditions. These conditions comprise second-order sufficient optimality conditions as well as regularity conditions on the control, which consists of a source condition and a condition on the active sets. In addition, we show that these conditions are necessary for convergence rates under certain conditions. We also consider sparse optimal control problems and derive regularization error estimates for them. Numerical experiments underline the theoretical findings.

Second-Order Necessary Conditions for Stochastic Optimal Control Problems

The main purpose of this paper is to present some of our recent results about the second-order necessary conditions for stochastic optimal controls with the control variable entering into both the drift and the diffusion terms. In particular, when the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established, whereas when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of the Pontryagin-type maximum principle. Unlike deterministic optimal control problems or stochastic optimal control problems with control-independent diffusions, there exist some essential difficulties in deriving the pointwise second-order necessary optimality conditions from the integral conditions when the controls act in the diffusion terms of the stochastic control systems. Some techniques from Malliavin calculus are employed to overcome these difficulties. Moreover, it is found that, in contrast to the first-order necessary conditions, the correction part of the solution to the second-order adjoint equation appears in the pointwise second-order necessary conditions whenever the diffusion term depends on the control variable, even if the control region is convex.

No-Gap Second-Order Conditions via a Directional Curvature Functional

This paper is concerned with necessary and sufficient second-order conditions for finite-dimensional and infinite-dimensional constrained optimization problems. Using a suitably defined directional curvature functional for the admissible set, we derive no-gap second-order optimality conditions in an abstract functional analytic setting. Our theory not only covers those cases where the classical assumptions of polyhedricity or second-order regularity are satisfied but also allows to study problems in the absence of these requirements. As a tangible example, we consider no-gap second-order conditions for bang-bang optimal control problems.

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

ABSTRACTAn optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.

Second-Order Necessary Conditions for a Strong Local Minimum in a Control Problem with General Control Constraints

We establish some second-order necessary optimality conditions for strong local minima in the Mayer type optimal control problem with a general control constraint $$U \subset \mathrm{I\! R}^m$$ and final state constraint described by a finite number of inequalities. In the difference with the main approaches of the existing literature, the second order tangents to U and the second order linearization of the control system are used to derive the second-order necessary conditions. This framework allows to replace the control system at hand by a differential inclusion with convex right-hand side. Such a relaxation of control system, and the differential calculus of derivatives of set-valued maps lead to fairly general statements. We illustrate the results by considering the case of control constraints defined by inequalities involving functions with positively or linearly independent gradients of active constraints, but also in the cases where the known approaches do not apply.

Euler discretization for a class of nonlinear optimal control problems with control appearing linearly

We investigate Euler discretization for a class of optimal control problems with a nonlinear cost functional of Mayer type, a nonlinear system equation with control appearing linearly and constraints defined by lower and upper bounds for the controls. Under the assumption that the cost functional satisfies a growth condition we prove for the discrete solutions Holder type error estimates w.r.t. the mesh size of the discretization. If a stronger second-order optimality condition is satisfied the order of convergence can be improved. Numerical experiments confirm the theoretical findings.

Numerical Optimal Control With Periodicity Constraints in the Presence of Invariants

Periodic optimal control problems (POCPs) based on dynamic models holding invariants are often problematic to treat using standard numerical methods. The difficulty stems from a failure of standard constraint qualifications and typically hinders the convergence of the numerical solver, or even defeats it. Optimization problems having weak constraint qualifications can be treated using dedicated solvers, at the price of a more involved algorithmic. In this paper, we analyze the constraint qualification of POCPs holding invariants, and propose three simple and computationally inexpensive modifications of the formulation that allow for a recovery of linear independence constraint qualification, while not affecting the second-order sufficient conditions for optimality. Hence, the resulting POCP can be tackled via standard solvers, without special treatment. The application of these approaches is detailed for the case of POCPs holding index-reduced differential-algebraic equations and representations of the $\mathrm{SO}\left(3\right)$ Lie group.

Variational Analysis of Nash Equilibrium

Tools of variational analysis are employed in studying the local stability of Nash equilibrium in a game-like framework of multi-agent optimization that emphasizes first- and second-order conditions for local optimality. The equilibrium is tied to a parameterized variational inequality for which properties of metric regularity can be derived.

Inference of Aircraft Intent via Inverse Optimal Control Including Second-Order Optimality Condition

This paper presents a method to infer the intent of an aircraft by using inverse optimal control. Assuming that the observed or estimated trajectory of the target aircraft is the approximate local optimum, the current method calculates the weight of each term of the objective function that represents the flight’s intent. The intent may include direct travel to a waypoint; circling around a waypoint; holding airspeed/heading/turn rate; avoiding certain regions such as those with adverse weather or special-use airspaces; and resolving conflicts with other aircraft. By incorporating the second-order optimality condition (i.e., the positive definiteness of the projected Hessian of the Lagrangian), the method approximately satisfies the sufficiency of the local optimality of the given trajectory. Moreover, the method also guarantees the uniqueness of the inferred intent. The effectiveness of the current method in terms of the reasonableness of the inferred intent as well as real-time applicability is demonstrated through numerical examples.

Restricted Robinson Constraint Qualification and Optimality for Cardinality-Constrained Cone Programming

In this paper, optimality conditions are presented and analyzed for the cardinality-constrained cone programming arising from finance, statistical regression, signal processing, etc. By introducing a restricted form of (strict) Robinson constraint qualification, the first-order optimality conditions for the cardinality-constrained cone programming are established based upon the properties of the normal cone. After characterizing further the second-order tangent set to the cardinality-constrained system, the second-order optimality conditions are also presented under some mild conditions. These proposed optimality conditions, to some extent, enrich the optimization theory for noncontinuous and nonconvex programming problems.

Second-order optimality conditions for nonlinear programs and mathematical programs

It is well known that second-order information is a basic tool notably in optimality conditions and numerical algorithms. In this work, we present a generalization of optimality conditions to strongly convex functions of order γ with the help of first- and second-order approximations derived from (Optimization 40(3):229-246, 2011) and we study their characterization. Further, we give an example of such a function that arises quite naturally in nonlinear analysis and optimization. An extension of Newton’s method is also given and proved to solve Euler equation with second-order approximation data.

Second-order optimality conditions for cone constrained multi-objective optimization

The aim of this paper is to develop second-order necessary and second-order sufficient optimality conditions for cone constrained multi-objective optimization. First of all, we derive, for an abstract constrained multi-objective optimization problem, two basic necessary optimality theorems for weak efficient solutions and a second-order sufficient optimality theorem for efficient solutions. Secondly, basing on the optimality results for the abstract problem, we demonstrate, for cone constrained multi-objective optimization problems, the first-order and second-order necessary optimality conditions under Robinson constraint qualification as well as the second-order sufficient optimality conditions under upper second-order regularity for the conic constraint. Finally, using the optimality conditions for cone constrained multi-objective optimization obtained, we establish optimality conditions for polyhedral cone, second-order cone and semi-definite cone constrained multi-objective optimization problems.

On the Rayleigh–Bénard–Marangoni system and a related optimal control problem

In this paper, we study a boundary control problem associated to the stationary Rayleigh–Bénard–Marangoni (RBM) system in presence of controls for the velocity and the temperature on parts of the boundary. We analyze the existence, uniqueness and regularity of weak solutions for the stationary RBM system in polyhedral domains of R3, and then, we prove the existence of the optimal solution. By using the Theorem of Lagrange multipliers, we derive an optimality system. We also give a second-order sufficient optimality condition and we establish a result of uniqueness of the optimal solution.