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

Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear or quadratic models due to the inability of currently available solvers to solve NLP problems of typical sizes. However stochastic programming problems are highly structured. The key to the efficient solution of such problems is therefore the ability to exploit their structure. Interior point methods are well-suited to the solution of very large non-linear optimization problems. In this paper we exploit this feature and show how portfolio optimization problems with sizes measured in millions of constraints and decision variables, featuring constraints on semi-variance, skewness or non-linear utility functions in the objective, can be solved with the state-of-the-art solver.     

A new efficient short-step projective interior point method for linear programming

In this paper, we are interested in the performance of Karmarkar’s projective algorithm for linear programming. We propose a new displacement step to accelerate and improve the convergence of this algorithm. This purpose is confirmed by numerical experimentations showing the efficiency and the robustness of the obtained algorithm over Schrijver’s one for small problem dimensions.     

An interior point method, based on rank-1 updates, for linear programming

We propose a polynomial time primal—dual potential reduction algorithm for linear programming. The algorithm generates sequencesd ^k andv ^k rather than a primal—dual interior point (x ^k ,s ^k ), where and fori = 1, 2,,n. Only one element ofd ^k is changed in each iteration, so that the work per iteration is bounded by O(mn) using rank-1 updating techniques. The usual primal—dual iteratesx ^k ands ^k are not needed explicitly in the algorithm, whereasd ^k andv ^k are iterated so that the interior primal—dual solutions can always be recovered by aforementioned relations between (x ^k, s^k) and (d ^k, v^k) with improving primal—dual potential function values. Moreover, no approximation ofd ^k is needed in the computation of projection directions. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization

This paper proves local convergence rates of primal-dual interior point methods for general nonlinearly constrained optimization problems. Conditions to be satisfied at a solution are those given by the usual Jacobian uniqueness conditions. Proofs about convergence rates are given for three kinds of step size rules. They are: (i) the step size rule adopted by Zhang et al. in their convergence analysis of a primal-dual interior point method for linear programs, in which they used single step size for primal and dual variables; (ii) the step size rule used in the software package OB1, which uses different step sizes for primal and dual variables; and (iii) the step size rule used by Yamashita for his globally convergent primal-dual interior point method for general constrained optimization problems, which also uses different step sizes for primal and dual variables. Conditions to the barrier parameter and parameters in step size rules are given for each case. For these step size rules, local and quadratic convergence of the Newton method and local and superlinear convergence of the quasi-Newton method are proved. A preliminary version of this paper was presented at the conference “Optimization-Models and Algorithms” held at the Institute of Statistical Mathematics, Tokyo, March 1993.     

Solving nested-constraint resource allocation problems with an interior point method

Efficient methods for convex resource allocation problems usually exploit algebraic properties of the objective function. For problems with nested constraints, we show that constraint sparsity structure alone allows rapid solution with a general interior point method. The key is a special-purpose linear system solver requiring only linear time in the problem dimensions. Computational tests show that this approach outperforms the previous best algebraically specialized methods.     

A strong bound on the integral of the central path curvature and its relationship with the iteration-complexity of primal-dual path-following LP algorithms

The main goals of this paper are to: i) relate two iteration-complexity bounds derived for the Mizuno-Todd-Ye predictor-corrector (MTY P-C) algorithm for linear programming (LP), and; ii) study the geometrical structure of the LP central path. The first iteration-complexity bound for the MTY P-C algorithm considered in this paper is expressed in terms of the integral of a certain curvature function over the traversed portion of the central path. The second iteration-complexity bound, derived recently by the authors using the notion of crossover events introduced by Vavasis and Ye, is expressed in terms of a scale-invariant condition number associated with m × n constraint matrix of the LP. In this paper, we establish a relationship between these bounds by showing that the first one can be majorized by the second one. We also establish a geometric result about the central path which gives a rigorous justification based on the curvature of the central path of a claim made by Vavasis and Ye, in view of the behavior of their layered least squares path following LP method, that the central path consists of long but straight continuous parts while the remaining curved part is relatively “short”. R. D. C. Monteiro was supported in part by NSF Grants CCR-0203113 and CCF-0430644 and ONR grant N00014-05-1-0183. T. Tsuchiya was supported in part by Japan-US Joint Research Projects of Japan Society for the Promotion of Science “Algorithms for linear programs over symmetric cones” and the Grants-in-Aid for Scientific Research (C) 15510144 of Japan Society for the Promotion of Science.     

A strongly polynomial simplex method for the linear fractional assignment problem

In this paper we show that the complexity of the simplex method for the linear fractional assignment problem (LFAP) is strongly polynomial. Although LFAP can be solved in polynomial time using various algorithms such as Newton’s method or binary search, no polynomial time bound for the simplex method for LFAP is known.     

A SCALED CENTRAL PATH FOR LINEAR PROGRAMMING

1. IntroductionIllterior poillt methods are one of the most illtensively studied topics in optinistion. Thousands of publications have been appeared on inferior point ndhods. A very good recent reviewis given by [4]. Interior point methods have very good theoretical properties including thenice polynomial complexity prOPerty. And more haportat is that numerous applications haveshown that interior point methods are very efficient for solving large sparse linear progr~ngproblemS. Interior poil…     

A primal-dual interior point method for parametric semidefinite programming problems

1.IntroductionSemidefiniteprogrammingunifiesquiteanumberofstandardmathematicalprogrammingproblems,suchaslinearprogrammingproblems,quadraticminimizationproblemswithconvexquadraticconstraints.Italsofindsmanyapplicationsinengineering,control,andcombinatorialoptimization[l,2].Inthepastfewyears,aquitenumberofresearchworkbasedoninteriorpointmethodsgaveattentiontoparametricsemidefiniteprogrammingproblems[3,4]fwherediscussionsaremostlyrelatedtopostoptimalandparametricanalysis.Inthispapergwefocusoureff…     

An adaptation of the dual-affine interior point method for the surface flatness problem

This paper presents an adaptation of the dual-affine interior point method for the surface flatness problem. In order to determine how flat a surface is, one should find two parallel planes so that the surface is between them and they are as close together as possible. This problem is equivalent to the problem of solving inconsistent linear systems in terms of Tchebyshev’s norm. An algorithm is proposed and results are presented and compared with others published in the literature.     

Solving the continuous nonlinear resource allocation problem with an interior point method

Resource allocation problems are usually solved with specialized methods exploiting their general sparsity and problem-specific algebraic structure. We show that the sparsity structure alone yields a closed-form Newton search direction for the generic primal-dual interior point method. Computational tests show that the interior point method consistently outperforms the best specialized methods when no additional algebraic structure is available.     

A primal projective interior point method for linear programming

We present a new projective interior point method for linear programming with unknown optimal value. This algorithm requires only that an interior feasible point be provided. It generates a strictly decreasing sequence of objective values and within polynomial time, either determines an optimal solution, or proves that the problem is unbounded. We also analyze the asymptotic convergence rate of our method and discuss its relationship to other polynomial time projective interior point methods and the affine scaling method.This research was supported in part by NSF Grants DMS-85-12277 and CDR-84-21402 and ONR Contract N00014-87-K0214.     

An active-set strategy in an interior point method for linear programming

We will present a potential reduction method for linear programming where only the constraints with relatively small dual slacks—termed active constraints—will be taken into account to form the ellipsoid constraint at each iteration of the process. The algorithm converges to the optimal feasible solution in O( L) iterations with the same polynomial bound as in the full constraints case, wheren is the number of variables andL is the data length. If a small portion of the constraints is active near the optimal solution, the computational cost to find the next direction of movement in one iteration may be considerably reduced by the proposed strategy.This research was partially done in June 1990 while the author was visiting the Department of Mathematics, University of Pisa.     

A primal—dual potential reduction method for problems involving matrix inequalities

We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worst-case analysis shows that the number of iterations grows as the square root of the problem size, but in practice it appears to grow more slowly. As in other interior-point methods the overall computational effort is therefore dominated by the least-squares system that must be solved in each iteration. A type of conjugate-gradient algorithm can be used for this purpose, which results in important savings for two reasons. First, it allows us to take advantage of the special structure the problems often have (e.g., Lyapunov or algebraic Riccati inequalities). Second, we show that the polynomial bound on the number of iterations remains valid even if the conjugate-gradient algorithm is not run until completion, which in practice can greatly reduce the computational effort per iteration.We describe in detail how the algorithm works for optimization problems withL Lyapunov inequalities, each of sizem. We prove an overallworst-case operation count of O(m ^5.5L^1.5). Theaverage-case complexity appears to be closer to O(m ⁴L^1.5). This estimate is justified by extensive numerical experimentation, and is consistent with other researchers' experience with the practical performance of interior-point algorithms for linear programming.This result means that the computational cost of extending current control theory based on the solution of Lyapunov or Riccatiequations to a theory that is based on the solution of (multiple, coupled) Lyapunov or Riccatiinequalities is modest.Supported by the Belgian National Fund for Scientific Research (NFWO). Research supported in part by the Belgian program on Interuniversity Attraction Poles (IUAP 17 and 50) initiated by the Belgian State, Prime Minister's Office, Science Policy Programming.Research supported in part by AFOSR (under F49620-92-J-0013), NSF (under ECS-9222391) and ARPA (under F49620-93-1-0085).     

A new polynomial time method for a linear complementarity problem

The purpose of this paper is to present a new polynomial time method for a linear complementarity problem with a positive semi-definite matrix. The method follows a sequence of points. If we generate the sequence on a path, we can construct a path following method, and if we generate the sequence based on a potential function, we can construct a potential reduction method. The method has the advantage that it requires at most iterations for any initial feasible point whose components lie between 2^–O(L) and 2^O(L).Research was supported in part by Grant-in-Aids for Encouragement of Young Scientists (63730014) and for General Scientific Research (63490010) of The Ministry of Education, Science and Culture.     

Improving the rate of convergence of interior point methods for linear programming

This paper proposes a procedure for improving the rate of convergence of interior point methods for linear programming. If (x ^k ) is the sequence generated by an interior point method, the procedure derives an auxiliary sequence ( ). Under the suitable assumptions it is shown that the sequence ( ) converges superlinearly faster to the solution than (x ^k ). Application of the procedure to the projective and afflne scaling algorithms is discussed and some computational illustration is provided.     

A general parametric analysis approach and its implication to sensitivity analysis in interior point methods

Adler and Monteiro (1992) developed a parametric analysis approach that is naturally related to the geometry of the linear program. This approach is based on the availability of primal and dual optimal solutions satisfying strong complementarity. In this paper, we develop an alternative geometric approach for parametric analysis which does not require the strong complementarity condition. This parametric analysis approach is used to develop range and marginal analysis techniques which are suitable for interior point methods. Two approaches are developed, namely the LU factorization approach and the affine scaling approach. Presented at the ORSA/TIMS, Nashville, TN, USA, May 1991. Supported by the National Science Foundation (NSF) under Grant No. DDM-9109404 and Grant No. DMI-9496178. This work was done while the author was a faculty member of the Systems and Industrial Engineering Department at The University of Arizona. Supported in part by the GTE Laboratories and the National Science Foundation (NSF) under Grant No. CCR-9019469.     

Warmstarting for interior point methods applied to the long-term power planning problem

The long-term planning of electricity generation in a liberalised market using the Bloom and Gallant model can be posed as a quadratic programming (QP) problem with an exponential number of linear inequality constraints called load-matching constraints (LMCs) and several other linear non-LMCs. Direct solution methods are inefficient at handling such problems and a heuristic procedure has been devised to generate only those LMCs that are likely to be active at the optimiser. The problem is then solved as a finite succession of QP problems with an increasing, though still limited, number of LMCs, which can be solved efficiently using a direct method, as would be the case with a QP interior-point algorithm. Warm starting between successive QP solutions helps then in reducing the number of iterations necessary to reach the optimiser.     

A primal interior point method for the linear semidefinite programming problem

The linear semidefinite programming problem is examined. A primal interior point method is proposed to solve this problem. It extends the barrier-projection method used for linear programs. The basic properties of the proposed method are discussed, and its local convergence is proved.     

A weighted least squaes study of robustness in interior point linear programming

This paper studies the robustness of interior point linear programming algorthims with respect to initial iterates that are too close to the boundary. Weighted least squares analysis is used in studying the near-boundary behavior of the affine scaling and Newton centering directions, which are often combined by interior point methods. This analysis leads to the develoment of a modified Newton centering direction exhibiting better near-boundary behavior than the two directions. Theoretical and computational results from the NETLIB test set are presented indicating that an approach which uses the modified newton direction is more robust than both the pure affine scaling approach and one which uses the Newton direction as the centering direction.