Second-order Conditions Research Articles

Optimality condition and complexity analysis for linearly-constrained optimization without differentiability on the boundary

In this paper we consider the minimization of a continuous function that is potentially not differentiable or not twice differentiable on the boundary of the feasible region. By exploiting an interior point technique, we present first- and second-order optimality conditions for this problem that reduces to classical ones when the derivative on the boundary is available. For this type of problems, existing necessary conditions often rely on the notion of subdifferential or become non-trivially weaker than the KKT condition in the (twice-)differentiable counterpart problems. In contrast, this paper presents a new set of first- and second-order necessary conditions that are derived without the use of subdifferential and reduces to exactly the KKT condition when (twice-)differentiability holds. As a result, these conditions are stronger than some existing ones considered for the discussed minimization problem when only non-negativity constraints are present. To solve for these optimality conditions in the special but important case of linearly constrained problems, we present two novel interior trust-region point algorithms and show that their worst-case computational efficiency in achieving the potentially stronger optimality conditions match the best known complexity bounds. Since this work considers a more general problem than the literature, our results also indicate that best known complexity bounds hold for a wider class of nonlinear programming problems.

Application of a Steady Meandering River with Piers Using a Lattice Boltzmann Sub-Grid Model in Curvilinear Coordinate Grid

A sub-grid multiple relaxation time (MRT) lattice Boltzmann model with curvilinear coordinates is applied to simulate an artificial meandering river. The method is based on the D2Q9 model and standard Smagorinsky sub-grid scale (SGS) model is introduced to simulate meandering flows. The interpolation supplemented lattice Boltzmann method (ISLBM) and the non-equilibrium extrapolation method are used for second-order accuracy and boundary conditions. The proposed model was validated by a meandering channel with a 180° bend and applied to a steady curved river with piers. Excellent agreement between the simulated results and previous computational and experimental data was found, showing that MRT-LBM (MRT lattice Boltzmann method) coupled with a Smagorinsky sub-grid scale (SGS) model in a curvilinear coordinates grid is capable of simulating practical meandering flows.

Analysis of Optimal Control Problems of Semilinear Elliptic Equations by BV-Functions

Optimal control problems for semilinear elliptic equations with control costs in the space of bounded variations are analysed. BV-based optimal controls favor piecewise constant, and hence ’simple’ controls, with few jumps. Existence of optimal controls, necessary and sufficient optimality conditions of first and second order are analysed. Special attention is paid on the effect of the choice of the vector norm in the definition of the BV-seminorm for the optimal primal and adjoined variables.

Second-Order Optimality Conditions for Multiobjective Optimization Whose Order Induced by Second-Order Cone

This paper is devoted to developing first-order necessary, second-order necessary, and second-order sufficient optimality conditions for a multiobjective optimization problem whose order is induced by a finite product of second-order cones (here named as Q-multiobjective optimization problem). For an abstract-constrained Q-multiobjective optimization problem, we derive two basic necessary optimality theorems for weak efficient solutions and a second-order sufficient optimality theorem for efficient solutions. For Q-multiobjective optimization problem with explicit constraints, we demonstrate first-order and second-order necessary optimality conditions under Robinson constraint qualification as well as second-order sufficient optimality conditions under upper second-order regularity for the explicit constraints. As applications, we obtain optimality conditions for polyhedral conic, second-order conic, and semi-definite conic Q-multiobjective optimization problems.

A second-order optimality condition with first- and second-order complementarity associated with global convergence of algorithms

We develop a new notion of second-order complementarity with respect to the tangent subspace related to second-order necessary optimality conditions by the introduction of so-called tangent multipliers. We prove that around a local minimizer, a second-order stationarity residual can be driven to zero while controlling the growth of Lagrange multipliers and tangent multipliers, which gives a new second-order optimality condition without constraint qualifications stronger than previous ones associated with global convergence of algorithms. We prove that second-order variants of augmented Lagrangian (under an additional smoothness assumption based on the Lojasiewicz inequality) and interior point methods generate sequences satisfying our optimality condition. We present also a companion minimal constraint qualification, weaker than the ones known for second-order methods, that ensures usual global convergence results to a classical second-order stationary point. Finally, our optimality condition naturally suggests a definition of second-order stationarity suitable for the computation of iteration complexity bounds and for the definition of stopping criteria.

A Test of Correlation in the Random Coefficients of an Autoregressive Process

A random coefficient autoregressive process in which the coefficients are correlated is investigated. First we look at the existence of a strictly stationary causal solution, we give the second-order stationarity conditions and the autocorrelation function of the process. Then we study some asymptotic properties of the empirical mean and the usual estimators of the process, such as convergence, asymptotic normality and rates of convergence, supplied with appropriate assumptions on the driving perturbations. Our objective is to get an overview of the influence of correlated coefficients in the estimation step through a simple model. In particular, the lack of consistency is shown for the estimation of the autoregressive parameter when the independence hypothesis in the random coefficients is violated. Finally, a consistent estimation is given together with a testing procedure for the existence of correlation in the coefficients. While convergence properties rely on ergodicity, we use a martingale approach to reach most of the results.

Improved local convergence results for augmented Lagrangian methods in $${\varvec{C}}^\mathbf{2}$$C2-cone reducible constrained optimization

This paper deals with a class of cone-reducible constrained optimization problems which encompasses nonlinear programming, semidefinite programming, second-order cone programming, and any combination thereof. Using the second-order sufficient condition and a strict version of the Robinson constraint qualification, we provide a (semi-)local error bound which generalizes known results from the literature. Moreover, under the same assumptions, we prove that an augmented Lagrangian method is locally convergent with rate proportional to $$1/\rho _k$$, where $$\rho _k$$ is the penalty parameter, and that $$\{\rho _k\}$$ remains bounded.

A numerical scheme to solve boundary value problems involving singular perturbation

In this study, a numerical method is presented in order to approximately solve singularly perturbed second-order differential equations givenwith boundary conditions. Themethod uses the set ofmonomialswhose degrees do not exceed a prescribed N as the set of base functions, resulting from the supposition that the approximate solution is a polynomial of degree N whose coefficients are to be determined. Then, following Galerkin’s approach, inner product with the base functions are applied to the residual of the approximate solution polynomial. This process, with a suitable incorporation of the boundary conditions, gives rise to an algebraic linear system of size N þ 1. The approximate polynomial solution is then obtained from the solution of this resulting system. Additionally, a technique, called residual correction,which exploits the linearity of the problem to estimate the error of any computed approximate solution is discussed briefly. The numerical scheme and the residual correction technique are illustrated with two examples.

Beyond Geometrical Optics: Higher Order Diffraction at Thin Curved Dielectric Layers

In this paper, we elaborate the method to successively find higher order terms of the asymptotic expansion of the field diffracted at a thin dielectric layer in 2-D problems (both the cases of E- and H-polarization). Each iteration of this method can be reduced to some standard 1-D boundary problem with a second-order differential equation and boundary conditions of impedance type. The boundary problem solvability and uniqueness are studied. The derivation of the leading asymptotic term is described, along with its geometrical optics meaning. The structure of the second asymptotic term is studied and analyzed with respect to its partial terms. Special attention is given to the partial terms describing the contribution to the diffracted field of the curvature of boundaries and their mutual slope. The results are verified by comparison with a solution obtained with the integral equation method.

On a Conjecture in Second-Order Optimality Conditions

In this paper, we deal with a conjecture formulated in Andreani et al. (Optimization 56:529–542, 2007), which states that whenever a local minimizer of a nonlinear optimization problem fulfills the Mangasarian–Fromovitz constraint qualification and the rank of the set of gradients of active constraints increases at most by one in a neighborhood of the minimizer, a second-order optimality condition that depends on one single Lagrange multiplier is satisfied. This conjecture generalizes previous results under a constant rank assumption or under a rank deficiency of at most one. We prove the conjecture under the additional assumption that the Jacobian matrix has a smooth singular value decomposition. Our proof also extends to the case of the strong second-order condition, defined in terms of the critical cone instead of the critical subspace.

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.

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.

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.

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.

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.

Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls

An optimal control problem for a semilinear elliptic equation of divergenceform is considered. Both the leading term and the semilinear term of the state equationcontain the control. The well-known Pontryagin type maximum principle for the optimal controls is the first-order necessary condition. When such a first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.

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.