Tuesday , October 23 2018

Optimization of a Constrained Quadratic Function

Charles HAMAKER
Department of Mathematics and Computer Science
PO Box 3517 Saint Mary’s College of California,
Moraga, CA,94575
chamaker@stmarys-ca.edu

Dedicated on the occasion of his 60th birthday to Neculai Andrei in appreciation of a lifetime of contributions to mathematics, as a productive researcher, an author of valuable texts and software, and a leader in the mathematical community.

Abstract: For A a positive definite, symmetric n x n matrix and b a real n-vector, the objective function Image826-2009,1,2 is optimized over the unit sphere. The proposed iterative methods, based on the gradient of f, converge in general for maximization and for large |b| for minimization with the principal computational cost being one or two matrix-vector multiplications per iteration. The rate of convergence improves as |b| increases, becoming computationally competitive in that case with algorithms developed for the more general problem wherein A may be indefinite.

Keywords: Constrained optimization, quadratic functions, iterated gradients, acceleration of convergence.

Charles Hamaker works in applicable analysis, particularly the application of Radon transforms to mathematical tomography and differential equations. He is also the director of his college’s core undergraduate program in reading classical texts of western culture.

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CITE THIS PAPER AS:
Charles HAMAKER, Optimization of a Constrained Quadratic Function, Studies in Informatics and Control, ISSN 1220-1766, vol. 18 (1), pp. 21-32, 2009.

1. Introduction and Preliminaries

Let A be a real-valued, positive definite, symmetric n x n matrix, and let b be a real-valued n-vector. The problem under consideration here is:

Optimize Image827-2009,1,2 over Image828-2009,1,2.

For small dimensions, the well-established approach involving diagonalization of A, outlined in [Golub and Van Loan 1996] and sketched in the remark preceding Lemma 1, suffices for both maximization and minimization. For large dimensions, the computational cost of diagonalization is excessive.

The maximization problem was encountered for n=4 in computations for lattice gauge simulations [Montero, 1999]. With A allowed to be indefinite, the minimization problem is the trust region subproblem which must be solved in each iteration of a trust region method [Conn, Gould, Toint, 2000] and [Hager, 2001]. The subproblem has been the subject of considerable attention, resulting in sophisticated iterative algorithms (see [Hager, 2001], [Sorenson, 1997], and [Rendl and Wolkowicz, 1997]). For instance, the SSM algorithm described in [Hager, 2001] converges quadratically, using the Lanczos process for startup and with each iterate being the solution of the problem on a low-dimensional subspace. That subspace is determined in part by applying Newton’s method at the previous iterate to the associated Lagrange problem. As is typical of these algorithms, the tradeoff for the small number of iterations required is that initialization and each iteration require a significant number of matrix-vector multiplications. See [Hager, 2001] for a comparison of the computational effectiveness of some of these algorithms. These minimization algorithms are adaptable to the maximization problem presumably with similar computational cost.

Denoting the eigenvalues of A by Image829-2009,1,2, direct the corresponding orthonormal basis of eigenvectors Image830-2009,1,2 so that

and in the case that Image832-2009,1,2, so that

and let E denote the orthogonal matrix having the  as columns. Then, since the change of coordinates  diagonalizes A and preserves the constraint set , it can be observed that

Thus, with respect to the coordinates , the problem takes the diagonalized form:

Optimize  over  where  and  are as above.

It is convenient to adopt the following notations. For , representation in coordinates will be with respect to the orthonormal eigenbasis  and will be denoted by . The outer subscript will be omitted when no ambiguity results, e.g.  while . The euclidean norm of a vector x will be denoted by |x|. The subset of consisting of all vectors with non-negative (non-positive) coordinates will be denoted by .

Geometric insight into the problem follows from completing the square in the diagonalized form of  to obtain

 where 

Thus, for c>-k, the level hypersurface f(x) = c is an ellipsoid with center at .

It is the simplicity of this geometric characterization that suggests the iterative algorithm for the maximization problem:  where T(x) is the normalization of the gradient. Note that the solution is a fixed point for T. While convergence in general will be established, it is reasonable to expect that the rate will improve as |b| increases, moving the common center of the ellipsoids further from the origin and thus resulting in less variation of  over the constraint set . The algorithm for the minimization problem (See section 4) is based on similar heuristic considerations.

Characterization of the maximizing vector  can begin by observing in the diagonalized form of the problem that it must have non-negative coordinates because the  are positive, and furthermore must satisfy the Lagrange multiplier condition:

 for some  (1)

Since  , this implies

 (2)

This leads to defining

and observing that the requirement that  means that μ must satisfy the secular equation g(t)=1.

If b1>0, then the requirement that  implies that , and hence that all the  are defined by (2). Because g(t) is unbounded and decreasing on  and approaches 0 as , there exists a unique  satisfying . Thus there is a unique maximizing point w.

In the case that  and , the requirement that  implies , and there is a unique  satisfying . Then (2) can be satisfied for i < k in one of two ways. First, if , then . Second, if  and  (in which case ) the unit vectors  determined by the Lagrange condition have coordinates  determined by (2) for , while . Thus , where

For  the unit vector determined by  in (2),, and  is positive for . Thus the desired maximum corresponds to the largest value of μ satisfying (1), namely the larger of  and .

 

Remark: Assuming that the problem has been diagonalized, i.e. the eigenvectors  have been found, and that , μ in equation (1) can be found by solving the secular equation g(t)=1 and then used in equation (2) to find w. This approach is outlined in [Golub and Van Loan, 1996] for the minimization problem and used in [Montero, 1999] to solve the maximization problem for n = 4.

The resulting characterization of the maximizing vector is summarized as follows.

Lemma 1

1. If , then the maximizing vector w satisfies  and the corresponding Lagrange multiplier satisfies .

2. If  and , then  and .

3. If  and , then  with the set S of maximizing vectors consisting of all w with  determined by (2) for  and satisfying .

Characterization of the minimizing vector v as corresponding via (2) to the smallest value  which satisfies (1) is completely similar, and is given in Lemma 2.2 of [Hager 2001]. In that paper, the counterparts to cases 1, 2, and 3. are referred to as non-degenerate, non-degenerate degenerate, and degenerate, and the literature on the trust region subproblem refers to cases 2 and 3 collectively as the hard case.

It is useful to characterize the optimizing vectors and their multipliers for |b| large.

Lemma 2 For , let b= τ u, τ > 0. Then, for fixed A,

1.  and .

2.  and 

Proof: The secular equation g(t) = 1 becomes

To establish case 1, note first that µo, the largest solution of the secular equation, satisfies . Thus for τ large enough,, and so . Setting t = μ in the secular equation, multiplying by , and taking limits gives

Then

 

 Case 2 is established similarly.

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