Second-order Conditions Research Articles

Experimental analysis of velocity slip at the wall for gas flows of nitrogen, R134a, and R600a through a metallic microtube

This paper reports measurements of mass flow rate of nitrogen, R134a, and R600a through a commercially available stainless steel microtube, which closely reproduces the case of gas leakage through micrometric clearances found in compressor valves. The accuracy of the measurements in the whole rarefaction range considered was verified through comparisons with predictions from the BGK kinetic model and results from the Navier–Stokes equations with modified boundary conditions. Furthermore, values of the tangential momentum accommodation coefficient (TMAC) were obtained by using an analytical approach derived from the solution of the Navier–Stokes equations with modified second-order boundary conditions. An incomplete accommodation is observed for all gases, with α = 0.933 ± 0.004 for nitrogen, 0.956 ± 0.004 for R134a, and 0.974 ± 0.008 for R600a. The results for all gases were very similar, indicating that the gas chemical composition does not significantly affect the flow through microtubes with rough internal surfaces, even when heavy polyatomic molecules are considered.

Globalizing Stabilized Sequential Quadratic Programming Method by Smooth Primal-Dual Exact Penalty Function

An iteration of the stabilized sequential quadratic programming method consists in solving a certain quadratic program in the primal-dual space, regularized in the dual variables. The advantage with respect to the classical sequential quadratic programming is that no constraint qualifications are required for fast local convergence (i.e., the problem can be degenerate). In particular, for equality-constrained problems, the superlinear rate of convergence is guaranteed under the only assumption that the primal-dual starting point is close enough to a stationary point and a noncritical Lagrange multiplier (the latter being weaker than the second-order sufficient optimality condition). However, unlike for the usual sequential quadratic programming method, designing natural globally convergent algorithms based on the stabilized version proved quite a challenge and, currently, there are very few proposals in this direction. For equality-constrained problems, we suggest to use for the task linesearch for the smooth two-parameter exact penalty function, which is the sum of the Lagrangian with squared penalizations of the violation of the constraints and of the violation of the Lagrangian stationarity with respect to primal variables. Reasonable global convergence properties are established. Moreover, we show that the globalized algorithm preserves the superlinear rate of the stabilized sequential quadratic programming method under the weak conditions mentioned above. We also present some numerical experiments on a set of degenerate test problems.

Optimal Control of the Instationary Three Dimensional Navier-Stokes-Voigt Equations

ABSTRACTWe study an optimal control problem with quadratic objective functional for the three dimensional Navier-Stokes-Voigt equations in bounded domains. We show the existence of optimal solutions, the necessary optimality conditions and the sufficient optimality conditions. The second-order optimality conditions obtained in the article seem to be optimal.

Characterization of quadratic growth of extended-real-valued functions

This paper shows that the sharpest possible bound in the second-order growth condition of a proper lower semicontinuous function can be attained under some assumptions. We also establish a relationship among strong metric subregularity, quadratic growth, the positive-definiteness property of the second-order subdifferential/generalized Hessian, the strong metric regularity, and tilt stability in a finite-dimensional setting.

A stabilized filter SQP algorithm for nonlinear programming

This paper presents a stabilized filter sequential quadratic programming (SQP) method for the general nonlinear optimization problems. The technique of stabilizing the inner quadratic programmings is an efficient strategy for the degenerate problem and brings the local superlinear convergence, while the integrated filter technique works effectively and guarantees the global convergence. The new algorithm works on both the primal and dual variables and solves the problem within the computational complexity comparable to the classical SQP algorithm. For the convergence, we show that (1) it converges either to a Karush---Kuhn---Tucker point at which the cone-continuity property holds, or to a stationary point in the sense of minimizing the constraint violation, and (2) under some second-order sufficient conditions, it converges locally superlinearly without any constraint qualifications. Our preliminary numerical results on a set of CUTEr test problems as well as on degenerate problems demonstrate the efficiency of the new algorithm.

Stochastic singular optimal control problem of switching systems with constraints

This paper is devoted to the optimal control problem of switching system in which constraints on the state variable are given by inclusions. Using Ekeland’s variational principle, second-order necessary condition of optimality for stochastic switching systems with constraints is obtained, and transversality conditions for switching law are established.

Computing Extreme Eigenvalues of Large Scale Hankel Tensors

Large scale tensors, including large scale Hankel tensors, have many applications in science and engineering. In this paper, we propose an inexact curvilinear search optimization method to compute Z- and H-eigenvalues of mth order n dimensional Hankel tensors, where n is large. Owing to the fast Fourier transform, the computational cost of each iteration of the new method is about $$\mathcal {O}(mn\log (mn))$$O(mnlog(mn)). Using the Cayley transform, we obtain an effective curvilinear search scheme. Then, we show that every limiting point of iterates generated by the new algorithm is an eigen-pair of Hankel tensors. Without the assumption of a second-order sufficient condition, we analyze the linear convergence rate of iterate sequence by the Kurdyka---źojasiewicz property. Finally, numerical experiments for Hankel tensors, whose dimension may up to one million, are reported to show the efficiency of the proposed curvilinear search method.

Second-Order Optimality Conditions for Boundary Control Problems with Mixed Pointwise Constraints

This paper deals with second-order optimality conditions for a boundary control problem which is governed by semilinear elliptic equations with mixed pointwise state-control constraints. We show that in some cases, there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, second-order sufficient optimality conditions for the problem where the objective function does not depend on control variables are also discussed.

Using necessary optimality conditions for acceleration of the nonuniform covering optimization method

Abstract Paper deals with the non-uniform covering method that is aimed at deterministic global optimization. This method finds a feasible solution to the optimization problem numerically and proves that the obtained solution differs from the optimal by no more than a given accuracy. Numerical proof consists of constructing a set of covering sets - the coverage. The number of elements in the coverage can be very large and even exceed the total amount of available computer resources. Basic method of coverage construction is the comparison of upper and lower bounds on the value of the objective function. In this work we propose to use necessary optimality conditions of first and second order for reducing the search for boxconstrained problems. We provide the algorithm description and prove its correctness. The efficiency of the proposed approach is studied on test problems.

Nonconvex Phase Synchronization

We estimate $n$ phases (angles) from noisy pairwise relative phase measurements. The task is modeled as a nonconvex least-squares optimization problem. It was recently shown that this problem can be solved in polynomial time via convex relaxation, under some conditions on the noise. In this paper, under similar but more restrictive conditions, we show that a modified version of the power method converges to the global optimum. This is simpler and (empirically) faster than convex approaches. Empirically, they both succeed in the same regime. Further analysis shows that, in the same noise regime as previously studied, second-order necessary optimality conditions for this quadratically constrained quadratic program are also sufficient, despite nonconvexity.

Generic Minimizing Behavior in Semialgebraic Optimization

We present a theorem of Sard type for semialgebraic set-valued mappings whose graphs have dimension no larger than that of their range space: the inverse of such a mapping admits a single-valued analytic localization around any pair in the graph, for a generic value parameter. This simple result yields a transparent and unified treatment of generic properties of semialgebraic optimization problems: “typical” semialgebraic problems have finitely many critical points, around each of which they admit a unique “active manifold” (analogue of an active set in nonlinear optimization); moreover, such critical points satisfy strict complementarity and second-order sufficient conditions for optimality are indeed necessary.

Nonrigorous symmetric second-order ABC applied to large-domain finite element modeling of electromagnetic scatterers

Nonrigorous symmetric second-order absorbing boundary condition (ABC) is presented as a feasible local mesh truncation in the higher-order large-domain finite element method (FEM) for electromagnetic analysis of scatterers in the frequency domain. The ABC is implemented on large generalized curvilinear hexahedral finite elements without imposing normal field continuity and without introducing new variables. As the extension of our previous work, the method is comprehensively evaluated by analyzing several benchmark targets, i.e., a metallic sphere, a dielectric cube, and NASA almond. Numerical examples show that radar cross section (RCS) of analyzed scatterers can be accurately predicted when the divergence term is included in computations nonrigorously. An influence of specific terms in the second-order ABC, which absorb transverse electric (TE) and transverse magnetic (TM) spherical modes, is also investigated. Examples show significant improvements in accuracy of the nonrigorous second-order ABC over the first-order ABC.

A class of new tail index estimators

In the paper, we propose a new class of functions which is used to construct tail index estimators. Functions from this new class are non-monotone in general, but they are the product of two monotone functions: the power function and the logarithmic function, which play essential role in the classical Hill estimator. The newly introduced generalized moment ratio estimator and generalized Hill estimator have a better asymptotic performance compared with the corresponding classical estimators over the whole range of the parameters that appear in the second-order regular variation condition. Asymptotic normality of the introduced estimators is proved, and comparison (using asymptotic mean square error) with other estimators of the tail index is provided. Some preliminary simulation results are presented.

Pointwise second-order necessary conditions for optimal control problems evolved on Riemannian manifolds

In this Note, we study an optimal control problem on a Riemannian manifold. The control set in our problem is assumed to be a general Polish space, and therefore the classical variation technique fails. We obtain a pointwise second-order optimality condition, for which the curvature tensor of the manifold appears explicitly in the second-order dual equation.

Zero-inflated compound Poisson distributions in integer-valued GARCH models

In this paper we introduce a wide class of integer-valued stochastic processes that allows to take into consideration, simultaneously, relevant characteristics observed in count data namely zero inflation, overdispersion and conditional heteroscedasticity. This class includes, in particular, the compound Poisson, the zero-inflated Poisson and the zero-inflated negative binomial INGARCH models, recently proposed in literature. The main probabilistic analysis of this class of processes is here developed. Precisely, first- and second-order stationarity conditions are derived, the autocorrelation function is deduced and the strict stationarity is established in a large subclass. We also analyse in a particular model the existence of higher-order moments and deduce the explicit form for the first four cumulants, as well as its skewness and kurtosis.

An inverse model and mathematical solution for inferring viscoelastic properties and dynamic deformations of heterogeneous structures

We proposed an adjoint-based inverse method and mathematical solutions using Lagrange multiplier theorem for inferring dynamic viscoelastic properties of heterogeneous structures with displacement observations to improve computation reliability and efficiency. Existing methods such as the one using finite-difference approximated gradient may not be efficient and accurate enough for considering complex situations of the coupled effects of dynamic loading, material deformation memory, and structural heterogeneity. The proposed method derives both gradient and Hessian mathematically to satisfy the first-order necessary and second-order sufficient optimal conditions, which has great advantages compared to other approaches. We also proposed robust numerical algorithms for accurate and fast computations, including a regularization to control reasonable parameter ranges, a Newton’s method with Hessian function to determine search direction for fast convenience, and a modified Armijo rule to find a stable step length. We developed a Galerkin time-domain finite-element method for numerical solutions of resulting partial differential equations, and validated the model for a layered structure under dynamic loading tests. Results indicate that the proposed method has greatly improved computation accuracy and speed as compared to existing approaches. It may be applied to a broad range of heterogeneous materials and structures at different length and time scales.

High-order maximum principles for the stability analysis of positive bilinear control systems

Summary We consider a continuous-time positive bilinear control system, which is a bilinear control system with Metzler matrices. The positive orthant is an invariant set of such a system, and the corresponding transition matrix is entrywise nonnegative for all time. Motivated by the stability analysis of positive linear switched systems under arbitrary switching laws, we define a control as optimal if it maximizes the spectral radius of the transition matrix at a given final time. We derive high-order necessary conditions for optimality for both singular and bang–bang controls. Our approach is based on combining results on the second-order derivative of a simple eigenvalue with the generalized Legendre-Clebsch condition and the Agrachev–Gamkrelidze second-order optimality condition. Copyright © 2015 John Wiley & Sons, Ltd.

Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

Second-order sufficient optimality conditions are considered for a simplified class of semilinear parabolic equations with quadratic objective functional including distributed and terminal observation. Main emphasis is laid on problems where the objective functional does not include a Tikhonov regularization term. Here, standard second-order conditions cannot be expected to hold. For this case, new second-order conditions are established that are based on different types of critical cones. Depending on the choice of this cones, the second-order conditions are sufficient for local minima that are weak or strong in the sense of calculus of variations.

First- and Second-Order Necessary Optimality Conditions for Optimal Control Problems Governed by Stationary Navier–Stokes Equations with Pure State Constraints

Based on tools of variational analysis and the diffuse variation method, we derive the Pontryagin maximum principle and second-order necessary optimality conditions for optimal control problems governed by stationary Navier–Stokes equations with pure state constraints. Our results are established under a smallness assumption on the control and the optimality conditions are of Fritz–John type.

On the Convergence Properties of a Second-Order Augmented Lagrangian Method for Nonlinear Programming Problems with Inequality Constraints

The objective of this paper is to conduct a theoretical study on the convergence properties of a second-order augmented Lagrangian method for solving nonlinear programming problems with both equality and inequality constraints. Specifically, we utilize a specially designed generalized Newton method to furnish the second-order iteration of the multipliers and show that when the linear independent constraint qualification and the strong second-order sufficient condition hold, the method employed in this paper is locally convergent and possesses a superlinear rate of convergence, although the penalty parameter is fixed and/or the strict complementarity fails.