Alexander D. Ioffe

Euler-Lagrange and Hamiltonian formalisms in dynamic optimization

We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler– Lagrange and Hamiltonian formalisms. The principal new results are: an extension of the recently discovered form of the Euler–Weierstrass condition to nonconvex valued differential inclusions, and a new Hamiltonian condition for convex valued inclusions. In both cases additional attention was given to weakening Lipschitz type requirements on the set–valued mapping. The central role of the Euler type condition is emphasized by showing that both the new Hamiltonian condition and the most general form of the Pontriagin maximum principle for equality constrained control systems are consequences of the Euler–Weierstrass condition. An example is given demonstrating that the new Hamiltonian condition is strictly stronger than the previously known one.

Variational Principles and Well-Posedness in Optimization and Calculus of Variations

The concluding result of the paper states that variational problems are generically solvable (and even well-posed in a strong sense) without the convexity and growth conditions always present in individual existence theorems. This and some other generic well-posedness theorems are obtained as realizations of a general variational principle extending the variational principle of Deville--Godefroy--Zizler.

Proximal analysis in smooth spaces

We provide a highly-refined sequential description of the generalized gradients of Clarke and approximate G-subdifferential of a lower semicontinuous extended-real-valued function defined on a Banach space with a β-smooth equivalent renorm. In the case of a Fréchet differentiable renorm, we give a corresponding result for the corresponding singular objects.

A Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional constraints

It is shown that a Lagrange multiplier rule involving the Michel-Penot subdifferentials is valid for the problem: minimizef0(x) subject tofi(x) ⩽ 0,i = 1, ⋯,m;fi(x) = 0,i = m + 1,⋯,n;x ∈Q where all functionsf are Lipschitz continuous andQ is a closed convex set. The proof is based on the theory of fans.

An Invitation to Tame Optimization

The word “tame” is used in the title in the same context as in expressions like “convex optimization,” “nonsmooth optimization,” etc.—as a reference to the class of objects involved in the formulation of optimization problems. Definable and tame functions and mappings associated with various o-minimal structures (e.g. semilinear, semialgebraic, globally subanalytic, and others) have a number of remarkable properties which make them an attractive domain for various applications. This relates both to the power of results that can be obtained and the power of available analytic techniques. The paper surveys certain ideas and recent results, some new, which have been or (hopefully) can be productively used in studies relating to variational analysis and nonsmooth optimization.

Second-order Sufficiency and Quadratic Growth for Nonisolated Minima

For standard nonlinear programming problems, the weak second-order sufficient condition is equivalent to the quadratic growth condition as far as the set of minima consists of isolated points and some qualification hypothesis holds. This kind of condition is instrumental in the study of numerical algorithms and sensitivity analysis. The arm of the paper is to study the relations between various types of sufficient conditions and quadratic growth in cases when the set of minima may have nonisolated points.

Variational analysis of a composite function: A formula for the lower second order epi-derivative☆

An exact formula is established for the lower second order epi-derivative of a function of the form g(F(x)), where F is a smooth map from one Banach space into another and g is a convex function (generally, not everywhere finite). Unconstrained minimization of such functions typically arise as an equivalent (in one or another sense) reduction form for many important classes of constrained optimization problems. The formula is further applied to study epi-differentiability of the max-function ƒ(x) = max{ƒ(q, x):q ϵ Q}.

On Sensitivity Analysis of Nonlinear Programs in Banach Spaces: The Approach via Composite Unconstrained Optimization

This article studies local sensitivity analysis of nonlinear programming problems in Banach spaces with arbitrary sets of solutions and no a priori regularity properties. This is done by first developing a corresponding theory for unconstrained optimization involving simple composite functions and then applying this theory to general nonlinear programs by way of certain reduction principles.

Towards variational analysis in metric spaces: metric regularity and fixed points

The main results of the paper include (a) a theorem containing estimates for the surjection modulus of a “partial composition” of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutin’s theorem about additive perturbation of a mapping into a Banach space by a Lipschitz mapping; (b) a “double fixed point” theorem for a couple of mappings, one from X into Y and another from Y to X which implies a fairly general version of the set-valued contraction mapping principle and also a certain (different) version of the first theorem.

Transversality and Alternating Projections for Nonconvex Sets

We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear convergence. When the two sets are semi-algebraic and bounded, but not necessarily transversal, we nonetheless prove subsequence convergence.