First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems

First-order and second-order necessary and sufficient optimality conditions are given for infinite-dimensional programming problems with constraints defined by arbitrary closed convex cones. The necessary conditions are immediate generalizations of those known for the finite-dimensional case. However, this does not hold for the sufficient conditions as illustrated by a counterexample. Here, to go from finite to infinite dimensions, causes an essential change in the proof-techniques and the results. We present modified sufficient conditions of first-order and of second-order which are based on a strengthening of the usual assumptions on the derivative of the objective function and on the second derivative of the Lagrangian.     

A Transformation Approach in Shape Optimization: Existence and Regularity Results

In this article, we consider a model shape optimization problem. The state variable solves an elliptic equation on a star-shaped domain, where the radius is given via a control function. First, we reformulate the problem on a fixed reference domain, where we focus on the regularity needed to ensure the existence of an optimal solution. Second, we introduce the Lagrangian and use it to show that the optimal solution possesses a higher regularity, which allows for the explicit computation of the derivative of the reduced cost functional as a boundary integral. We finish the article with some second-order optimality conditions.     

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.     

Reciprocal Diffusions and Symmetries of Parabolic PDE: The Nonflat Case

We introduce a new set of Reciprocal Characteristics for the class of reciprocal diffusions naturally associated to a general parabolic second-order linear differential operator. All the coefficients of this operator, including the diffusion matrix, depend on time. This set of reciprocal characteristics is provided by the study of the symmetries of the differential operator. The Riemannian metric defined by the diffusion matrix is of central importance. Our reciprocal characteristics are the natural extension of Ovsiannikov's differential invariants to the time dependent parabolic case. We also show that the symmetries of the PDE coincide with the one parameter families of transformations which leave the usual stochastic Lagrangian as well as a modified Onsager–Machlup Lagrangian invariant.     

Augmentability in optimization theory

The present paper is concerned with the concept of augmentability. One form of augmentability is based on the generalized Lagrangian and leads to the method of multipliers for solving constrained minimum problems. An assumption of augmentability can be used in place of regularity to establish first-order and second-order necessary conditions for a minimum for constrained minimum problems. Another type of augmentability leads to the theory of Mayer fields in variational theory.Dedicated to R. Bellman     

Quasimodes and Bohr-Sommerfeld Conditions for the Toeplitz Operators

Abstract

This article is devoted to the quantization of the Lagrangian submanifolds in the context of geometric quantization. The objects we define are similar to the Lagrangian distributions of the cotangent phase space theory. We apply this to construct quasimodes for the Toeplitz operators and we state the Bohr-Sommerfeld conditions under the usual regularity assumption. To compare with the Bohr-Sommerfeld conditions for a pseudodifferential operator with small parameter, the Maslov index, defined from the vertical polarization, is replaced with a curvature integral, defined from the complex polarization. We also consider the quantization of the symplectomorphisms, the realization of semi-classical equivalence between two different quantizations of a symplectic manifold and the microlocal equivalences.     

Noether, partial Noether operators and first integrals for a linear system

We obtain Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a system of two linear second-order ordinary differential equations (ODEs) with variable coefficients. The canonical form for a system of two second-order ordinary differential equations is invoked and a special case of this system is studied for both Noether and partial Noether operators. Then the first integrals with respect to Noether and partial Noether operators are obtained for the linear system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. This can give rise to further studies on systems from a partial Lagrangian viewpoint as systems in general do not admit Lagrangians.     

A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects

In this paper, we present a novel Lagrangian formulation of the equations of motion for point vortices on the unit 2-sphere. We show first that no linear Lagrangian formulation exists directly on the 2-sphere but that a Lagrangian may be constructed by pulling back the dynamics to the 3-sphere by means of the Hopf fibration. We then use the isomorphism of the 3-sphere with the Lie group SU(2) to derive a variational Lie group integrator for point vortices which is symplectic, second-order, and preserves the unit-length constraint. At the end of the paper, we compare our integrator with classical fourth-order Runge–Kutta, the second-order midpoint method, and a standard Lie group Munthe-Kaas method.     

Marginal values and second-order necessary conditions for optimality

Second-order necessary conditions in nonlinear programming are derived by a new method that does not require the usual sort of constraint qualification. In terms of the multiplier vectors appearing in such second-order conditions, an estimate, is obtained for the generalized subgradients of the optimal, value function associated with a parameterized nonlinear programming problem. This yields estimates for ‘marginal values’ with respect to the parameters. The main theoretical tools are the augmented Lagrangian and, despite the assumption of second-order smoothness of objective constraints, the subdifferential calculus that has recently been developed for nonsmooth, nonconvex functions. Research sponsored in part by the Air Force Office of Scientific, Research, Air Force Systems Command United States Air Force, under grant No. F4960-82-k-0012.     

Perturbation analysis of short-crested waves in Lagrangian coordinates

A new second-order asymptotic solution that describes short-crested waves is derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid satisfies the irrotational condition and there being zero pressure at the free surface, in contrast with the Eulerian solution, in which there is residual pressure at the free surface. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level. The mass transport velocity and Lagrangian mean level associated with particle displacement can also be obtained directly. In particular, the mean level of the particle motion in a Lagrangian form differs that of the Eulerian form. The new formulation reduces to second-order standing or progressive wave solutions in Lagrangian coordinates at the limiting angles of approach. Expressions for kinematic quantities are also presented.     

Extended convergence results for the method of multipliers for nonstrictly binding inequality constraints

In this paper, we extend the classical convergence and rate of convergence results for the method of multipliers for equality constrained problems to general inequality constrained problems, without assuming the strict complementarity hypothesis at the local optimal solution. Instead, we consider an alternative second-order sufficient condition for a strict local minimum, which coincides with the standard one in the case of strict complementary slackness. As a consequence, new stopping rules are derived in order to guarantee a local linear rate of convergence for the method, even if the current Lagrangian is only asymptotically minimized in this more general setting. These extended results allow us to broaden the scope of applicability of the method of multipliers, in order to cover all those problems admitting loosely binding constraints at some optimal solution. This fact is not meaningless, since in practice this kind of problem seems to be more the rule rather than the exception.In proving the different results, we follow the classical primaldual approach to the method of multipliers, considering the approximate minimizers for the original augmented Lagrangian as the exact solutions for some adequate approximate augmented Lagrangian. In particular, we prove a general uniform continuity property concerning both their primal and their dual optimal solution set maps, a property that could be useful beyond the scope of this paper. This approach leads to very simple proofs of the preliminary results and to a straight-forward proof of the main results.The author gratefully acknowledges the referees for their helpful comments and remarks. This research was supported by FONDECYT (Fondo Nacional de Desarrollo Científico y Technológico de Chile).     

Hamiltonian structure of the Yang–Mills functional

Hamilton equations based upon a general Lepagean equivalent of the Yang–Mills Lagrangian are investigated. A regularization of the Yang–Mills Lagrangian which is singular with respect to the standard regularity conditions is derived.     

Stationary Points of Bound Constrained Minimization Reformulations of Complementarity Problems

We consider two merit functions which can be used for solving the nonlinear complementarity problem via nonnegatively constrained minimization. One of the functions is the restricted implicit Lagrangian (Refs. 1–3), and the other appears to be new. We study the conditions under which a stationary point of the minimization problem is guaranteed to be a solution of the underlying complementarity problem. It appears that, for both formulations, the same regularity condition is needed. This condition is closely related to the one used in Ref. 4 for unrestricted implicit Lagrangian. Some new sufficient conditions are also given.     

The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones

The focus of this paper is on studying an inverse second-order cone quadratic programming problem, in which the parameters in the objective function need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem with cone constraints, and its dual, which has fewer variables than the original one, is a semismoothly differentiable (SC ¹) convex programming problem with both a linear inequality constraint and a linear second-order cone constraint. We demonstrate the global convergence of the augmented Lagrangian method with an exact solution to the subproblem and prove that the convergence rate of primal iterates, generated by the augmented Lagrangian method, is proportional to 1/r, and the rate of multiplier iterates is proportional to $1/\sqrt{r}$ , where r is the penalty parameter in the augmented Lagrangian. Furthermore, a semismooth Newton method with Armijo line search is constructed to solve the subproblems in the augmented Lagrangian approach. Finally, numerical results are reported to show the effectiveness of the augmented Lagrangian method with both an exact solution and an inexact solution to the subproblem for solving the inverse second-order cone quadratic programming problem.     

Second-order negative-curvature methods for box-constrained and general constrained optimization

A Nonlinear Programming algorithm that converges to second-order stationary points is introduced in this paper. The main tool is a second-order negative-curvature method for box-constrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (Powell-Hestenes-Rockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to second-order stationary points in situations in which first-order methods fail are exhibited.     

Local convergence of filter methods for equality constrained non-linear programming

In Gonzaga et al. [A globally convergent filter method for nonlinear programming, SIAM J. Optimiz. 14 (2003), pp. 646–669] we discuss general conditions to ensure global convergence of inexact restoration filter algorithms for non-linear programming. In this article we show how to avoid the Maratos effect by means of a second-order correction. The algorithms are based on feasibility and optimality phases, which can be either independent or not. The optimality phase differs from the original one only when a full Newton step for the tangential minimization of the Lagrangian is efficient but not acceptable by the filter method. In this case a second-order corrector step tries to produce an acceptable point keeping the efficiency of the rejected step. The resulting point is tested by trust region criteria. Under the usual hypotheses, the algorithm inherits the quadratic convergence properties of the feasibility and optimality phases. This article includes a comparison between classical Sequential Quadratic Programming (SQP) and Inexact Restoration (IR) iterations, showing that both methods share the same asymptotic convergence properties.     

The multiplier method of Hestenes and Powell applied to convex programming

For nonlinear programming problems with equality constraints, Hestenes and Powell have independently proposed a dual method of solution in which squares of the constraint functions are added as penalties to the Lagrangian, and a certain simple rule is used for updating the Lagrange multipliers after each cycle. Powell has essentially shown that the rate of convergence is linear if one starts with a sufficiently high penalty factor and sufficiently near to a local solution satisfying the usual second-order sufficient conditions for optimality. This paper furnishes the corresponding method for inequality-constrained problems. Global convergence to an optimal solution is established in the convex case for an arbitrary penalty factor and without the requirement that an exact minimum be calculated at each cycle. Furthermore, the Lagrange multipliers are shown to converge, even though the optimal multipliers may not be unique.This work was supported in part by the Air Force Office of Scientific Research under Grant No. AF-AFOSR-72-2269.     

The regularity of solutions of reaction-diffusion equations via Lagrangian coordinates

In this paper we study the regularity of nonnegative solutions and their interfaces for the nonlinear reaction-diffusion equation $$u_t = \left( {u^m } \right)_{xx} + f\left( u \right),\left( E \right)$$ wherem>1 andf(u) is aC ¹ function withf(0)=0 and is subject to some other technical conditions. This equation has the property of finite propagation which gives rise to interfaces separating regions whereu=0 andu>0. The analysis is carried out by means of Lagrangian coordinates, formally viewing the reaction-diffusion equation as the equation governing the evolution of the density of a certain continuum. Lagrangian coordinates have been successfully applied to study nonlinear diffusion equations posed in one space dimension. The usual formulation applies to equations which can be written in the form of a conservation law, which is not the case here because of the reaction term. In problems exhibiting interfaces such technique has the merit of rendering the interfaces straight lines, much simplifying the analysis. In this paper we present anon-standard Lagrangian formulation that works innon-conservation cases. Equation (E) is then translated into this framework and we find in a natural way the necessary estimates to prove theC ¹ regularity of moving interfaces and the regularity of the weak solution near such an interface, that allows us to establish the dynamic properties of the interface for the solutions. We end the paper by describing how the method can be applied to similar problems inseveral space dimensions with radial symmetry.     

Augmented Lagrangian method for second-order cone programs under second-order sufficiency

This paper addresses problems of second-order cone programming important in optimization theory and applications. The main attention is paid to the augmented Lagrangian method (ALM) for such problems considered in both exact and inexact forms. Using generalized differential tools of second-order variational analysis, we formulate the corresponding version of second-order sufficiency and use it to establish, among other results, the uniform second-order growth condition for the augmented Lagrangian. The latter allows us to justify the solvability of subproblems in the ALM and to prove the linear primal–dual convergence of this method.