Larry D. Irvine

Bio: Larry D. Irvine is an academic researcher. The author has contributed to research in topics: Smoothing spline & Nonlinear programming. The author has an hindex of 1, co-authored 1 publications receiving 80 citations.

Constrained interpolation and smoothing

Abstract: Numerical and theoretical questions related to constrained interpolation and smoothing are treated. The prototype problem is that of finding the smoothest convex interpolant to given univariate data. Recent results have shown that this convex programming problem with infinite constraints can be recast as a finite parametric nonlinear system whose solution is closely related to the second derivative of the desired interpolating function. This paper focuses on the analysis of numerical techniques for solving the nonlinear system and on the theoretical issues that arise when certain extensions of the problem are considered. In particular, we show that two standard iteration techniques, the Jacobi and Gauss-Seidel methods, are globally convergent when applied to this problem. In addition we use the problem structure to develop an efficient implementation of Newton's method and observe consistent quadratic convergence. We also develop a theory for the existence, uniqueness, and representation of solutions to the convex interpolation problem with nonzero lower bounds on the second derivative (strict convexity). Finally, a smoothing spline analogue to the convex interpolation problem is studied with reference to the computation of convex approximations to noisy data.

Partially finite convex programming, Part I: Quasi relative interiors and duality theory

Abstract: We study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. We use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable.

A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix

Abstract: The nearest correlation matrix problem is to find a correlation matrix which is closest to a given symmetric matrix in the Frobenius norm. The well-studied dual approach is to reformulate this problem as an unconstrained continuously differentiable convex optimization problem. Gradient methods and quasi-Newton methods such as BFGS have been used directly to obtain globally convergent methods. Since the objective function in the dual approach is not twice continuously differentiable, these methods converge at best linearly. In this paper, we investigate a Newton-type method for the nearest correlation matrix problem. Based on recent developments on strongly semismooth matrix valued functions, we prove the quadratic convergence of the proposed Newton method. Numerical experiments confirm the fast convergence and the high efficiency of the method.

Duality relationships for entropy-like minimization problems

Abstract: This paper considers the minimization of a convex integral functional over the positive cone of an $L_p $ space, subject to a finite number of linear equality constraints. Such problems arise in spectral estimation, where the bjective function is often entropy-like, and in constrained approximation. The Lagrangian dual problem is finite-dimensional and unconstrained. Under a quasi-interior constraint qualification, the primal and dual values are equal, with dual attainment. Examples show the primal value may not be attained. Conditions are given that ensure that the primal optimal solution can be calculated directly from a dual optimum. These conditions are satisfied in many examples.

A General Projection Framework for Constrained Smoothing

Abstract: There are a wide arrayof smoothing methods available for finding structure in data. A general framework is developed which shows that manyof these can be viewed as a projection of the data, with respect to appropriate norms. The underlying vector space is an unusually large product space, which allows inclusion of a wide range of smoothers in our setup (including manymethods not typicallyconsidered to be projec- tions). We give several applications of this simple geometric interpreta- tion of smoothing. A major payoff is the natural and computationally frugal incorporation of constraints. Our point of view also motivates new estimates and helps understand the finite sample and asymptotic behavior of these estimates.