On the projected subgradient method for nonsmooth convex optimization in a Hilbert space

We consider the method for constrained convex optimization in a Hilbert space, consisting of a step in the direction opposite to an _k -subgradient of the objective at a current iterate, followed by an orthogonal projection onto the feasible set. The normalized stepsizes _k are exogenously given, satisfying _{k=0 k} = , _{k=0 k} ² < , and _k is chosen so that _{k k} for some > 0. We prove that the sequence generated in this way is weakly convergent to a minimizer if the problem has solutions, and is unbounded otherwise. Among the features of our convergence analysis, we mention that it covers the nonsmooth case, in the sense that we make no assumption of differentiability off, and much less of Lipschitz continuity of its gradient. Also, we prove weak convergence of the whole sequence, rather than just boundedness of the sequence and optimality of its weak accumulation points, thus improving over all previously known convergence results. We present also convergence rate results. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research of this author was partially supported by CNPq grant nos. 301280/86 and 300734/95-6.     

A class of bounds for convex bodies in Hilbert space

We extend to infinite dimensions a class of bounds forL _p metrics of finite-dimensional convex bodies. A generalization to arbitrary increasing convex functions is done simultaneously. The main tool is the use of Gaussian measure to effect a normalization for varying dimension. At a point in the proof we also invoke a strong law of large numbers for random sets to produce a rotational averaging.Supported in part by ONR Grant N0014-90-J-1641 and NSF Grant DMS-9002665.     

The method of cyclic projections for closed convex sets in a Hilbert space under the presence of computational errors

Numerical Algorithms - In this paper, we study the method of cyclic projections for inconsistent convex feasibility problems in a Hilbert space under the presence of computational errors. We show...     

Broyden's method in Hilbert space

Broyden's method is formulated for the solution of nonlinear operator equations in Hilbert spaces. The algorithm is proven to be well defined and a linear rate of convergence is shown. Under an additional assumption on the initial approximation for the derivative we prove the superlinear rate of convergence.     

A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems

This paper presents a new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between our method and other methods for solving an open fourth-order boundary value problem presented by Scott and Watts. The method is also applied to a nonlinear fourth-order boundary value problem. The numerical results demonstrate that the new method is quite accurate and efficient for fourth-order boundary value problems.     

Stable sequential convex programming in a Hilbert space and its application for solving unstable problems

A parametric convex programming problem with an operator equality constraint and a finite set of functional inequality constraints is considered in a Hilbert space. The instability of this problem and, as a consequence, the instability of the classical Lagrange principle for it is closely related to its regularity and the subdifferentiability properties of the value function in the optimization problem. A sequential Lagrange principle in nondifferential form is proved for the indicated convex programming problem. The principle is stable with respect to errors in the initial data and covers the normal, regular, and abnormal cases of the problem and the case where the classical Lagrange principle does not hold. It is shown that the classical Lagrange principle in this problem can be naturally treated as a limiting variant of its stable sequential counterpart. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimal control problems and inverse problems is discussed. For two illustrative problems of these kinds, the corresponding stable Lagrange principles are formulated in sequential form.     

A method for estimating interpolating polynomials

The question of obtaining a lower bound for some interpolating polynomials is considered. Under specific conditions it is proved that these bounds are sharp. As a corollary of the general theorem, under specific restrictions on the points of interpolation, lower bounds for Goncharov interpolation polynomials are obtained which coincide with known upper bounds.Translated from Matematicheskie Zametki, Vol. 17, No. 4, pp. 555–561, April, 1975.     

The number of weakly compact convex subsets of the Hilbert space

We prove that for κ an uncountable cardinal, there exist κ² many nonhomeomorphic weakly compact convex subsets of weight κ in the Hilbert space ?₂(κ).     

An extraproximal method for solving equilibrium programming problems in a Hilbert space

A regularized method of the proximal type for solving equilibrium problems in a Hilbert space is proposed. The method is combined with an approximation of the original problem. The convergence of the method is analyzed, and a regularizing operator is constructed.     

An alternating projections method for certain linear problems in a Hilbert space

Recent extensions of von Neumann's alternating projection methodpermit the computation of proximity projections onto certainconvex sets. This paper exploits this fact in constructing aglobally convergent method for minimizing linear functions overa convex set in a Hilbert space. In particular, we solve theeducational testing problem and an inverse eigenvalue problem,two difficult problems involving positive semidefiniteness constraints.     

A new proof of an inequality of Bohr for Hilbert space operators

In this note we present a new proof and an extension of the Hilbert space operators version of an inequality by Bohr.