Interior-point Methods For Semidefinite Programming Research Articles

A primal–dual regularized interior-point method for semidefinite programming

Interior-point methods in semidefinite programming (SDP) require the solution of a sequence of linear systems which are used to derive the search directions. Safeguards are typically required in order to handle rank-deficient Jacobians and free variables. We generalize the primal–dual regularization of Friedlander and Orban [A primal–dual regularized interior-point method for convex quadratic programs, Math. Program. Comput. 4 (2012), pp. 71–107] to SDP and show that it is possible to recover an optimal solution of the original primal–dual pair by taking one step of Newton method to a sequence of regularized SDPs at each iteration for both the Nesterov–Todd and dual Helmberg–Kojima–Monteiro (HKM) directions. Computationally, a sparse factorization may be used on a sparse augmented system instead of the more costly symmetric indefinite factorization. Benefits of our approach include increased robustness and a simpler implementation. Our method does not require the constraints to be linearly independent and does not assume that Slater's condition holds. We report numerical experience on standard problems that illustrate our findings.

Projection methods for quantum channel construction

We consider the problem of constructing quantum channels, if they exist, that transform a given set of quantum states $$\{\rho _1, \ldots , \rho _k\}$${?1,?,?k} to another such set $$\{\hat{\rho }_1, \ldots , \hat{\rho }_k\}$${?^1,?,?^k}. In other words, we must find a completely positive linear map, if it exists, that maps a given set of density matrices to another given set of density matrices, possibly of different dimension. Using the theory of completely positive linear maps, one can formulate the problem as an instance of a positive semidefinite feasibility problem with highly structured constraints. The nature of the constraints makes projection-based algorithms very appealing when the number of variables is huge and standard interior-point methods for semidefinite programming are not applicable. We provide empirical evidence to this effect. We moreover present heuristics for finding both high-rank and low-rank solutions. Our experiments are based on the method of alternating projections and the Douglas---Rachford reflection method.

Semi-definite programming-based method for security-constrained unit commitment with operational and optimal power flow constraints

Considering the economics and securities for the operation of a power system, a semi-definite programming (SDP) model for the security-constrained unit commitment (SCUC) problem is described here, which is directly solved by the interior-point method for SDP within the polynomial times. The proposed method is promising for the SCUC problems because of its excellent convergence and the ability of handling the non-covex integer variables. No model decomposition and initial relaxation are needed when applying the SDP-based method. When the solution contains minor mismatches in the integer variables, a simple rounding strategy is used to correct the non-integer into integer efficiently. Different test cases from 6 to 118 buses over a 24 h horizon are presented. Extensive numerical simulations have shown that the proposed method is capable of obtaining the optimal UC schedules without any network and bus voltage violations, and minimising the operation cost as well.

The [formula omitted] method for second order cone programming

We develop the Q method for the second order cone programming problem. The algorithm is the adaptation of the Q method for semidefinite programming originally developed by Alizadeh, Haeberly and Overton [A new primal–dual interior point method for semidefinite programming. In: Proceedings of the fifth SIAM conference on applications of linear algebra, Snowbird, Utah, 1994.] and [Primal–dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results. SIAM Journal on Optimization 1998;8(3):746–68 [electronic].]. We take advantage of the special algebraic structure associated with second order cone programs to formulate the Q method. Furthermore we discuss the convergence properties of the algorithm. Finally, some numerical results are presented.

Convex optimization problems involving finite autocorrelation sequences

We discuss convex optimization problems in which some of the variables are constrained to be finite autocorrelation sequences. Problems of this form arise in signal processing and communications, and we describe applications in filter design and system identification. Autocorrelation constraints in optimization problems are often approximated by sampling the corresponding power spectral density, which results in a set of linear inequalities. They can also be cast as linear matrix inequalities via the Kalman-Yakubovich-Popov lemma. The linear matrix inequality formulation is exact, and results in convex optimization problems that can be solved using interior-point methods for semidefinite programming. However, it has an important drawback: to represent an autocorrelation sequence of length $n$, it requires the introduction of a large number ($n(n+1)/2$) of auxiliary variables. This results in a high computational cost when general-purpose semidefinite programming solvers are used. We present a more efficient implementation based on duality and on interior-point methods for convex problems with generalized linear inequalities.

A Note on the Calculation of Step-Lengths in Interior-Point Methods for Semidefinite Programming

In each iteration of an interior-point method for semidefinite programming, the maximum step-length that can be taken by the iterate while maintaining the positive semidefiniteness constraint needs to be estimated. In this note, we show how the maximum step-length can be estimated via the Lanczos iteration, a standard iterative method for estimating the extremal eigenvalues of a matrix. We also give a posteriori error bounds for the estimate. Numerical results on the performance of the proposed method against two commonly used methods for calculating step-lengths (backtracking via Cholesky factorizations and exact eigenvalues computations) are included.

Solving Some Large Scale Semidefinite Programs via the Conjugate Residual Method

Most current implementations of interior-point methods for semidefinite programming use a direct method to solve the Schur complement equation (SCE) $M \Delta y=h$ in computing the search direction. When the number of constraints is large, the problem of having insufficient memory to store M can be avoided if an iterative method is used instead. Numerical experiments have shown that the conjugate residual (CR) method typically takes a huge number of steps to generate a high accuracy solution. On the other hand, it is difficult to incorporate traditional preconditioners into the SCE, except for block diagonal preconditioners. We decompose the SCE into a 2 × 2 block system by decomposing $\Delta y$ (similarly for h) into two orthogonal components with one lying in a certain subspace that is determined from the structure of M. Numerical experiments on semidefinite programming problems arising from the Lovász $\theta$-function of graphs and MAXCUT problems show that high accuracy solutions can be obtained with a moderate number of CR steps using the proposed equation.

Primal-Dual Interior-Point Methods for Semidefinite Programming in Finite Precision

Recently, a number of primal-dual interior-point methods for semidefinite programming have been developed. To reduce the number of floating point operations, each iteration of these methods typically performs block Gaussian elimination with block pivots that are close to singular near the optimal solution. As a result, these methods often exhibit complex numerical properties in practice. We consider numerical issues related to some of these methods. Our error analysis indicates that these methods could be numerically stable if certain coefficient matrices associated with the iterations are well-conditioned, but are unstable otherwise. With this result, we explain why one particular method, the one introduced by Alizadeh, Haeberly, and Overton is in general more stable than others. We also explain why the so-called least squares variation, introduced for some of these methods, does not yield more numerical accuracy in general. Finally, we present results from our numerical experiments to support our analysis.

A study of search directions in primal-dual interior-point methods for semidefinite programming*

We discuss several different search directions which can be used in primal-dual interior-point methods for semidefinite programming problems and investigate their theoretical properties, including scale invariance, primal-dual symmetry, and whether they always generate well-defined directions. Among the directions satisfying all but at most two of these desirable properties are the Alizadeh-Haeberly-Overton, Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-HaralMonteiro, Nesterov-Todd, Gu, and Toh directions, as well as directions we will call the MTW and Half directions. The first five of these appear to be the best in our limited computational testing also.

A unified approach to statistical design centering of integrated circuits with correlated parameters

This paper presents a general method for statistical design centering of integrated circuits with correlated parameters. It unifies worst-case design, nominal design and tolerance design in a single framework by selecting appropriate norms to measure the distances from the nominal values. The method uses an advanced first-order second moment technique as an alternative to the simplicial algorithm. Yield estimation is calculated in the original space and no transformation to uncorrelated variables is required. The solution algorithms are based on the recently developed interior-point methods for semi-definite programming. One tutorial and one practical example explain the application.

A note on the Nesterov-Todd and the Kojima-Shindoh-hara search directions in semidefinite programming

This short note shows that the Nesterov-Todd search direction used in primal-dual interior-point methods for semidefinite programs belongs to the family of search directions proposed by Kojima, Shindoh and Hara.

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 Newton's 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.

On Extending Some Primal--Dual Interior-Point Algorithms From Linear Programming to Semidefinite Programming

This work concerns primal--dual interior-point methods for semidefinite programming (SDP) that use a search direction originally proposed by Helmberg et al. [SIAM J. Optim., 6 (1996), pp. 342--361] and Kojima, Shindoh, and Hara [SIAM J. Optim., 7 (1997), pp. 86--125.] and recently rediscovered by Monteiro [SIAM J. Optim., 7 (1997), pp. 663--678] in a more explicit form. In analyzing these methods, a number of basic equalities and inequalities were developed in [Kojima, Shindoh, and Hara] and also in [Monteiro] through different means and in different forms. In this paper, we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming (LP), producing basic relationships that have compact forms almost identical to their counterparts in LP. We also introduce a new formulation of the central path and variable-metric measures of centrality. These results provide convenient tools for deriving polynomiality results for primal--dual algorithms extended from LP to SDP using the aforementioned and related search directions. We present examples of such extensions, including the long-step infeasible-interior-point algorithm of Zhang [SIAM J. Optim., 4 (1994), pp. 208--227].

Reconstruction with Noisy Data: An Approach via Eigenvalue Optimization

We present a nonlinear inverse filtering approach to problems such as power spectrum estimation of stationary time series or deconvolution of a blurred image. The technique is based on eigenvalue optimization and a numerical treatment may therefore be obtained using primal-dual interior-point methods for semidefinite programming.

Optimizing dominant time constant in RC circuits

Conventional methods for optimal sizing of wires and transistors use linear resistor-capacitor (RC) circuit models and the Elmore delay as a measure of signal delay. If the RC circuit has a tree topology, the sizing problem reduces to a convex optimization problem that can be solved using geometric programming. The tree topology restriction precludes the use of these methods in several sizing problems of significant importance to high-performance deep submicron design, including for example, circuits with loops of resistors, e.g., clock distribution meshes and circuits with coupling capacitors, e.g., buses with crosstalk between the wires. In this paper, we propose a new optimization method that can be used to address these problems. The method is based on the dominant time constant as a measure of signal propagation delay in an RC circuit instead of Elmore delay. Using this measure, sizing of any RC circuit can be cast as a convex optimization problem and solved using recently developed efficient interior-point methods for semidefinite programming. The method is applied to three important sizing problems: clerk mesh sizing and topology design, sizing of tristate buses, and sizing of bus line widths and spacings taking crosstalk into account.

Exploiting sparsity in primal-dual interior-point methods for semidefinite programming

The Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro and Nesterov-Todd search directions have been used in many primal-dual interior-point methods for semidefinite programs. This paper proposes an efficient method for computing the two directions when the semidefinite program to be solved is large scale and sparse.