Michael J. Best

Active set algorithms for isotonic regression: a unifying framework

AbstractIn this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods. The active set approach provides a unifying framework for studying algorithms for isotonic regression, simplifies the exposition of existing algorithms and leads to several new efficient algorithms. We also investigate the computational complexity of several algorithms.In this paper we consider the isotonic regression problem with respect to a complete order

An Algorithm for the Solution of the Parametric Quadratic Programming Problem

\begin{gathered} minimize\sum\limits_{i = 1}^n {w_i } (y_i - x_i )^2 \hfill \\ subject tox_1 \leqslant x_2 \leqslant \cdot \cdot \cdot \leqslant x_n \hfill \\ \end{gathered}

Positively Weighted Minimum-Variance Portfolios and the Structure of Asset Expected Returns

where eachwi is strictly positive and eachyi is an arbitrary real number. We show that the Pool Adjacent Violators algorithm (due to Ayer et al., 1955; Miles, 1959; Kruskal, 1964), is a dual feasible active set method and that the Minimum Lower Set algorithm (due to Brunk et al., 1957) is a primal feasible active set method of computational complexity O(n2). We present a new O(n) primal feasible active set algorithm. Finally we discuss Van Eedens method and show that it is of worst-case exponential time complexity.

A feasible conjugate-direction method to solve linearly constrained minimization problems

Abstract. This paper develops a closed form solution of the mean-variance portfolio selection problem for uncorrelated and bounded assets when an additional technical assumption is satisfied. Although the assumption of uncorrelated assets is unduly restrictive, the explicit determination of the efficient asset holdings in the presence of bound constraints gives insight into the nature of the efficient frontier. The mean-variance portfolio selection problem considered here deals with the budget constraint and lower bounds or the budget constraint and upper bounds. For the mean-variance portfolio selection problem dealing with lower bounds the closed form solution is derived for two cases: a universe of only risky assets and a universe of risky assets plus an additional asset which is risk free. For the mean-variance portfolio selection problem dealing with upper bounds, the results presented are for a universe consisting only of risky assets. In each case, the order in which the assets are driven to their bounds depends on the ordering of their expected returns.

An algorithm for portfolio optimization with transaction costs

We present an ”active set” algorithm for the solution of the convex (but not necessarily strictly convex) parametric quadratic programming problem. The optimal solution and associated multipliers are obtained as piece-wise linear functions of the parameter. At the end of each interval, the active set is changed by either adding, deleting, or exchanging a constraint. The method terminates when either the optimal solution has been obtained for all values of the parameter, or, a further increase in the parameter results in either the feasible region being null or the objective function being unbounded from below. The method used to solve the linear equations associated with a particular active set is left unspecified. The parametric algorithm can thus be implemented using the linear equation solving method of any active set quadratic programming algorithm.

Portfolio Selection and Transactions Costs

We formulate a general algorithm for the solution of a convex (but not strictly convex) quadratic programming problem. Conditions are given under which the iterates of the algorithm are uniquely determined. The quadratic programming algorithms of Fletcher, Gill and Murray, Best and Ritter, and van de Panne and Whinston/Dantzig are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points. The various methods are shown to differ only in the manner in which they solve the linear equations expressing the Kuhn-Tucker system for the associated equality constrained subproblems. Equivalence results have been established by Goldfarb and Djang for the positive definite Hessian case. Our analysis extends these results to the positive semi-definite case.

A globally and quadratically convergent algorithm for general nonlinear programming problems

In this paper, we derive simple, directly computable conditions for minimum-variance portfolios to have all positive weights. We show that either there is no minimum-variance portfolio with all positive weights or there is a single segment of the minimum-variance frontier for which all portfolios have positive weights. Then, we examine the likelihood of observing positively weighted minimum-variance portfolios. Analytical and computational results suggest that: i) even if the mean vector and covariance matrix are compatible with a given positively weighted portfolio being mean-variance efficient, the proportion of the minimum-variance frontier containing positively weighted portfolios is small and decreases as the number of assets in the universe increases, and ii) small perturbations in the means will likely lead to no positively weighted minimum-variance portfolios.

A Class of Accelerated Conjugate Direction Methods for Linearly Constrained Minimization Problems

An iterative procedure is presented which uses conjugate directions to minimize a nonlinear function subject to linear inequality constraints. The method (i) converges to a stationary point assuming only first-order differentiability, (ii) has ann-q step superlinear or quadratic rate of convergence with stronger assumptions (n is the number of variables,q is the number of constraints which are binding at the optimum), (iii) requires the computation of only the objective function and its first derivatives, and (iv) is experimentally competitive with well-known methods.