Parallel interior-point solver for structured quadratic programs: Application to financial planning problems

Abstract

Many practical large-scale optimization problems are not only sparse, but also display some form of block-structure such as primal or dual block angular structure. Often these structures are nested: each block of the coarse top level structure is block-structured itself. Problems with these characteristics appear frequently in stochastic programming but also in other areas such as telecommunication network modelling.We present a linear algebra library tailored for problems with such structure that is used inside an interior point solver for convex quadratic programming problems. Due to its object-oriented design it can be used to exploit virtually any nested block structure arising in practical problems, eliminating the need for highly specialised linear algebra modules needing to be written for every type of problem separately. Through a careful implementation we achieve almost automatic parallelisation of the linear algebra.The efficiency of the approach is illustrated on several problems arising in the financial planning, namely in the asset and liability management. The problems are modelled as multistage decision processes and by nature lead to nested block-structured problems. By taking the variance of the random variables into account the problems become non-separable quadratic programs. A reformulation of the problem is proposed which reduces density of matrices involved and by these means significantly simplifies its solution by an interior point method. The object-oriented parallel solver achieves high efficiency by careful exploitation of the block sparsity of these problems. As a result a problem with over 50 million decision variables is solved in just over 2 hours on a parallel computer with 16 processors. The approach is by nature scalable and the parallel implementation achieves nearly perfect speedups on a range of problems.

Solving non-linear portfolio optimization problems with the primal-dual interior point method

“…Indeed, all its constraints are linear and its objective is a concave quadratic function of x. The variance (4) gathers all x h T,j (assets held in the last time period) and thus produces a quadratic form x T Qx with a very large number of nonzero entries in matrix Q. Gondzio and Grothey [6] proposed a reformulation of the problem which exploits partial separability of the variance and leads to a non-convex formulation of the problem but with a much higher degree of sparsity in the quadratic term. The reformulation exploits another representation of the variance (using…”

“…• A constraint on downside risk (measured by the semi-variance): (6). (11) • A logarithmic utility function as the objective: (6).…”

“…(11) • A logarithmic utility function as the objective: (6). (12) • An objective including skewness: Konno et al [11] suggest using the third moment in the model to capture the investors preference towards positive deviations in the case of non-symmetric distribution of returns for some assets (2), (3), (6).…”

“…To solve the nonlinear ALM variations (10) through (13) we use a textbook sequential quadratic programming (SQP) method that employs our Object-Oriented Parallel interior point Solver (OOPS) [7,6] as the underlying QP solver. We have implemented an SQP procedure following [3].…”

“…Linear algebra approaches such as [2,19] are usually limited to very special structures resulting for example from constraints on the allowed type of recurrence relation. The aim of this paper is to show that by using a modern general structure exploiting interior point method such as our Object-Oriented Parallel Solver OOPS [7,6] the solution of very large, general nonlinear stochastic programs is feasible. Furthermore this method is not limited to a special setting but can be applied to almost any conceivable constraint or objective function in stochastic programming.…”