Farid Alizadeh

Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization

This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First the classical cone duality is reviewed as it is specialized to SDP is reviewed. Next an interior point algorithm is presented that converges to the optimal solution in polynomial time. The approach is a direct extension of Ye’s projective method for linear programming. It is also argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion. Finally, the significance of these results is studied in a variety of combinatorial optimization problems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graph...

Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results

Primal-dual interior-point path-following methods for semidefinite programming are considered. Several variants are discussed, based on Newtons method applied to three equations: primal feasibility, dual feasibility, and some form of centering condition. The focus is on three such algorithms, called the XZ, XZ+ZX, and Q methods. For the XZ+ZX and Q algorithms, the Newton system is well defined and its Jacobian is nonsingular at the solution, under nondegeneracy assumptions. The associated Schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption. Practical aspects are discussed, including Mehrotra predictor-corrector variants and issues of numerical stability. Compared to the other methods considered, the XZ+ZX method is more robust with respect to its ability to step close to the boundary, converges more rapidly, and achieves higher accuracy.

Extension of primal-dual interior point algorithms to symmetric cones

Abstract. In this paper we show that the so-called commutative class of primal-dual interior point algorithms which were designed by Monteiro and Zhang for semidefinite programming extends word-for-word to optimization problems over all symmetric cones. The machinery of Euclidean Jordan algebras is used to carry out this extension. Unlike some non-commutative algorithms such as the XS+SX method, this class of extensions does not use concepts outside of the Euclidean Jordan algebras. In particular no assumption is made about representability of the underlying Jordan algebra. As a special case, we prove polynomial iteration complexities for variants of the short-, semi-long-, and long-step path-following algorithms using the Nesterov-Todd, XS, or SX directions.

Complementarity and nondegeneracy in semidefinite programming

Primal and dual nondegeneracy conditions are defined for semidefinite programming. Given the existence of primal and dual solutions, it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution. The converses hold if strict complementarity is assumed. Primal and dual nondegeneracy assumptions do not imply strict complementarity, as they do in LP. The primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutionsX andZ. This is in contrast with LP where nondegeneracy assumptions exactly determine the number of variables which are zero. It is shown that primal and dual nondegeneracy and strict complementarity all hold generically. Numerical experiments suggest probability distributions for the ranks ofX andZ which are consistent with the nondegeneracy conditions.

Physical mapping of chromosomes: a combinatorial problem in molecular biology

This paper is concerned wth the physical mapping of DNA molecules using data about the hybridization of oligonucleotide probes to a library of clones. In mathematical terms, the DNA molecule corresponds to an interval on the real line, each clone to a subinterval, and each probe occurs at a finite set of points within the interval. A stochastic model for the occurrences of the probes and the locations of the clones is assumed. Given a matrix of incidences between probes and clones, the task is to reconstruct the most likely interleaving of the clones. Combinatorial algorithms are presented for solving approximations to this problem, and computational results are presented.

Associative and Jordan Algebras, and Polynomial Time Interior-Point Algorithms for Symmetric Cones

We present a general framework whereby analysis of interior-point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras. In particular, such analyses are extendible to the cone of positive semidefinite Hermitian matrices with complex and quaternion entries, and to the Lorentz cone. We prove the case of the Lorentz cone by using the embedding of its associated Jordan algebra in the Clifford algebra. As an example of such extensions we take Monterios polynomial-time complexity analysis of the family of similarly scaled directions--introduced by Monteiro and Zhang (1998)--and generalize it to cone-LP over all representable symmetric cones.

Physical mapping of chromosomes using unique probes

The goal of physical mapping of the genome is to reconstruct a strand of DNA given a collection of overlapping fragments, or clones, from the strand. We present several algorithms to infer how the clones overlap, given data about each clone. We focus on data used to map human chromosomes 21 and Y, in which relatively short substrings, or probes, are extracted from the ends of clones. The substrings are long enough to be unique with high probability. The data we are given is an incidence matrix of clones and probes. In the absence of error, the correct placement can be found easily using a PQ-tree. The data are never free from error, however, and algorithms are differentiated by their performance in the presence of errors. We approach errors from two angles: by detecting and removing them, and by using algorithms that are robust in the presence of errors. We have also developed a strategy to recover noiseless data through an interactive process that detects anomalies in the data and retests questionable entries in the incidence matrix of clones and probes. We evaluate the effectiveness of our algorithms empirically, using simulated data as well as real data from human chromosome 21.

Symmetric Cones, Potential Reduction Methods and Word-by-Word Extensions

This is the first of three chapters in this book dealing with polynomial time complexity of interior point algorithms for semidefinite programming (SDP). As such it deals with, in a sense, the easiest class of algorithms for which polynomial time convergence can be established. More precisely, we present a “recipe” whereby polynomial time convergence proofs in linear programming (LP) can be extended “word-by-word” to analogous proofs in SDP. Our presentation closely follows [21].

Shape-Constrained Estimation Using Nonnegative Splines

We consider the problem of nonparametric estimation of unknown smooth functions in the presence of restrictions on the shape of the estimator and on its support using polynomial splines. We provide a general computational framework that treats these estimation problems in a unified manner, without the limitations of the existing methods. Applications of our approach include computing optimal spline estimators for regression, density estimation, and arrival rate estimation problems in the presence of various shape constraints. Our approach can also handle multiple simultaneous shape constraints. The approach is based on a characterization of nonnegative polynomials that leads to semidefinite programming (SDP) and second-order cone programming (SOCP) formulations of the problems. These formulations extend and generalize a number of previous approaches in the literature, including those with piecewise linear and B-spline estimators. We also consider a simpler approach in which nonnegative splines are approximated by splines whose pieces are polynomials with nonnegative coefficients in a nonnegative basis. A condition is presented to test whether a given nonnegative basis gives rise to a spline cone that is dense in the space of nonnegative continuous functions. The optimization models formulated in the article are solvable with minimal running time using off-the-shelf software. We provide numerical illustrations for density estimation and regression problems. These examples show that the proposed approach requires minimal computational time, and that the estimators obtained using our approach often match and frequently outperform kernel methods and spline smoothing without shape constraints. Supplementary materials for this article are provided online.

Arrival Rate Approximation by Nonnegative Cubic Splines

We describe an optimization method to approximate the arrival rate of data such as e-mail messages, Web site visits, changes to databases, and changes to Web sites mirrored by other servers. We model these arrival rates as non-homogeneous Poisson process based on observed arrival data. We estimate the arrival function by cubic splines using the maximum likelihood principle. A critical feature of the model is that the splines are constrained to be everywhere nonnegative. We formulate this constraint using a characterization of nonnegative polynomials by positive semidefinite matrices. We also describe versions of our model that allow for periodic arrival rate functions and input data of limited precision. We formulate the estimation problem as a convex program related to semidefinite programming and solve it with a standard nonlinear optimization package called KNITRO. We present numerical results using both an actual record of e-mail arrivals over a period of sixty weeks, and artificially generated data sets. We also present a cross-validation procedure for determining an appropriate number of spline knots to model a set of arrival observations