Associative and Jordan Algebras, and Polynomial Time Interior-Point Algorithms for Symmetric Cones (original) (raw)

Associative algebras, symmetric cones and polynomial time interior point algorithms

1998

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 Monterio's polynomialtime 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.

Semidefinite Characterization of Sum-of-Squares Cones in Algebras

SIAM Journal on Optimization, 2013

We extend Nesterov's semidefinite programming characterization of squared functional systems, and Faybusovich's abstraction to bilinear symmetric maps, to cones of sum-of-squares elements in general abstract algebras. Using algebraic techniques such as isomorphism, linear isomorphism, tensor products, sums, and direct sums, we show that many concrete cones are in fact sum-of-squares cones with respect to some algebra and thus are representable by the cone of positive semidefinite matrices. We also consider nonnegativity with respect to a proper cone K and show that in some cases cones of K-nonnegative functions are either sum of squares or at least semidefinite representable. For example, we show that some well-known Chebyshev systems, when extended to Euclidean Jordan algebras, induce cones that are semidefinite representable. Finally we will discuss some concrete examples and applications, including minimum ellipsoid enclosing given space curves, minimization of eigenvalues of polynomial matrix pencils, approximation of functions by shapeconstrained functions, and approximation of combinatorial optimization problems by polynomial programming.

Intrinsic volumes of symmetric cones and applications in convex programming

Mathematical Programming, 2014

We express the probability distribution of the solution of a random (standard Gaussian) instance of a convex cone program in terms of the intrinsic volumes and curvature measures of the reference cone. We then compute the intrinsic volumes of the cone of positive semidefinite matrices over the real numbers, over the complex numbers, and over the quaternions in terms of integrals related to Mehta's integral. In particular, we obtain a closed formula for the probability that the solution of a random (standard Gaussian) semidefinite program has a certain rank.

Conic optimization with spectral functions on Euclidean Jordan algebras

2021

Spectral functions on Euclidean Jordan algebras arise frequently in convex models. Despite the success of primal-dual conic interior point solvers, there has been little work on enabling direct support for spectral cones, i.e. proper nonsymmetric cones defined from epigraphs and perspectives of spectral functions. We propose simple logarithmically homogeneous barriers for spectral cones and we derive efficient, numerically stable procedures for evaluating barrier oracles such as inverse Hessian operators. For two useful classes of spectral cones the root-determinant cones and the matrix monotone derivative cones we show that the barriers are self-concordant, with nearly optimal parameters. We implement these cones and oracles in our open source solver Hypatia, and we write simple, natural formulations for four applied problems. Our computational benchmarks demonstrate that Hypatia often solves the natural formulations more efficiently than advanced solvers such as MOSEK 9 solve equi...

Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets

SIAM Journal on Optimization, 2000

Let F be a compact subset of the n-dimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets C k (k = 1, 2,. . .) of R n such that (a) the convex hull of F ⊆ C k+1 ⊆ C k (monotonicity), (b) ∩ ∞ k=1 C k = the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding Lovász-Schrijver lift-and-project procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semiinfinite convex QOP relaxation proposed originally by Fujie and Kojima. Using this equivalence, we investigate some fundamental features of the two methods including (a) and (b) above.

The Simplest Semidefinite Programs are Trivial

Mathematics of Operations Research, 1995

We consider optimization problems of the following type: [Formula: see text] Here, tr(·) denotes the trace operator, C and X are symmetric n × n matrices, B is a symmetric m × m matrix and A(·) denotes a linear operator. Such problems are called semidefinite programs and have recently become the object of considerable interest due to important connections with max-min eigenvalue problems and with new bounds for integer programming. In the context of symmetric matrices, the simplest linear operators have the following form: [Formula: see text] where M is an arbitrary m × n matrix. In this paper, we show that for such linear operators the optimization problem is trivial in the sense that an explicit solution can be given.

An extension of Chubanov’s algorithm to symmetric cones

Mathematical Programming

In this work we present an extension of Chubanov's algorithm to the case of homogeneous feasibility problems over a symmetric cone K. As in Chubanov's method for linear feasibility problems, the algorithm consists of a basic procedure and a step where the solutions are confined to the intersection of a half-space and K. Following an earlier work by Kitahara and Tsuchiya on second order cone feasibility problems, progress is measured through the volumes of those intersections: when they become sufficiently small, we know it is time to stop. We never have to explicitly compute the volumes, it is only necessary to keep track of the reductions between iterations. We show this is enough to obtain concrete upper bounds to the minimum eigenvalues of a scaled version of the original feasibility problem. Another distinguishing feature of our approach is the usage of a spectral norm that takes into account the way that K is decomposed as simple cones. In several key cases, including semidefinite programming and second order cone programming, these norms make it possible to obtain better complexity bounds for the basic procedure when compared to a recent approach by Peña and Soheili. Finally, in the appendix, we present a translation of the algorithm to the homogeneous feasibility problem in semidefinite programming.

A Polynomial Primal-Dual Path-Following Algorithm for Second-order Cone Programming

Second-order cone programming (SOCP) is the problem of minimizing linear objective function over cross-section of second-order cones and an ane space. Recently this problem gets more attention because of its various important applications including quadratically constrained convex quadratic programming. In this paper we deal with a primal-dual path-following algorithm for SOCP to show many of the ideas developed for primal-dual algorithms for LP and SDP carry over to this problem. We dene neighborhoods of the central trajectory in terms of the \eigenvalues" of the second-order cone, and develop an analogue of HRVW/KSH/M direction, and establish O( p n log " 01 ), O(n log " 01 ) and O(n 3 log " 01 ) iteration-complexity bounds for short-step, semilong-step and long-step path-following algorithms, respectively, to reduce the duality gap by a factor of ". keywords: second-order cone, interior-point methods, polynomial complexity, primal-dual path-following methods.