A Full Nesterov–Todd Step Infeasible Interior-Point Method for Second-Order Cone Optimization

After a brief introduction to Jordan algebras, we present a primal–dual interior-point algorithm for second-order conic optimization that uses full Nesterov–Todd steps; no line searches are required. The number of iterations of the algorithm coincides with the currently best iteration bound for second-order conic optimization. We also generalize an infeasible interior-point method for linear optimization to second-order conic optimization. As usual for infeasible interior-point methods, the starting point depends on a positive number. The algorithm either finds a solution in a finite number of iterations or determines that the primal–dual problem pair has no optimal solution with vanishing duality gap.     

Curvature integrals and iteration complexities in SDP and symmetric cone programs

In this paper, we study iteration complexities of Mizuno-Todd-Ye predictor-corrector (MTY-PC) algorithms in SDP and symmetric cone programs by way of curvature integrals. The curvature integral is defined along the central path, reflecting the geometric structure of the central path. Integrating curvature along the central path, we obtain a precise estimate of the number of iterations to solve the problem. It has been shown for LP that the number of iterations is asymptotically precisely estimated with the integral divided by $\sqrt{\beta}$ , where β is the opening parameter of the neighborhood of the central path in MTY-PC algorithms. Furthermore, this estimate agrees quite well with the observed number of iterations of the algorithm even when β is close to one and when applied to solve large LP instances from NETLIB. The purpose of this paper is to develop direct extensions of these two results to SDP and symmetric cone programs. More specifically, we give concrete formulas for curvature integrals in SDP and symmetric cone programs and give asymptotic estimates for iteration complexities. Through numerical experiments with large SDP instances from SDPLIB, we demonstrate that the number of iterations is explained quite well with the integral even for a large step size which is enough to solve practical large problems.     

Central Swaths

We develop a natural generalization to the notion of the central path, a concept that lies at the heart of interior-point methods for convex optimization. The generalization is accomplished via the “derivative cones” of a “hyperbolicity cone”, the derivatives being direct and mathematically appealing relaxations of the underlying (hyperbolic) conic constraint, be it the non-negative orthant, the cone of positive semidefinite matrices, or other. We prove that a dynamics inherent to the derivative cones generates paths always leading to optimality, the central path arising from a special case in which the derivative cones are quadratic. Derivative cones of higher degree better fit the underlying conic constraint, raising the prospect that the paths they generate lead to optimality quicker than the central path.     

A homogeneous interior-point algorithm for nonsymmetric convex conic optimization

A homogeneous interior-point algorithm for solving nonsymmetric convex conic optimization problems is presented. Starting each iteration from the vicinity of the central path, the method steps in the approximate tangent direction and then applies a correction phase to locate the next well-centered primal–dual point. Features of the algorithm include that it makes use only of the primal barrier function, that it is able to detect infeasibilities in the problem and that no phase-I method is needed. We prove convergence to \(\epsilon \)-accuracy in \({\mathcal {O}}(\sqrt{\nu } \log {(1/\epsilon )})\) iterations. To improve performance, the algorithm employs a new Runge–Kutta type second order search direction suitable for the general nonsymmetric conic problem. Moreover, quasi-Newton updating is used to reduce the number of factorizations needed, implemented so that data sparsity can still be exploited. Extensive and promising computational results are presented for the \(p\)-cone problem, the facility location problem, entropy maximization problems and geometric programs; all formulated as nonsymmetric convex conic optimization problems.     

Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results

 In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two different ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix completion itself in a primal-dual interior-point method. The current article presents the details of their implementations. We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one. Numerical results over different classes of SDPs show that these methods can be very efficient for some problems. Received: March 18, 2001 / Accepted: May 31, 2001 Published online: October 9, 2002 RID="⋆" ID="⋆"The author was supported by The Ministry of Education, Culture, Sports, Science and Technology of Japan. Key Words. semidefinite programming – primal-dual interior-point method – matrix completion problem – clique tree – numerical results Mathematics Subject Classification (2000): 90C22, 90C51, 05C50, 05C05     

Cuts for mixed 0-1 conic programming

In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.     

Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones

We describe an implementation of nonsymmetric interior-point methods for linear cone programs defined by two types of matrix cones: the cone of positive semidefinite matrices with a given chordal sparsity pattern and its dual cone, the cone of chordal sparse matrices that have a positive semidefinite completion. The implementation takes advantage of fast recursive algorithms for evaluating the function values and derivatives of the logarithmic barrier functions for these cones. We present experimental results of two implementations, one of which is based on an augmented system approach, and a comparison with publicly available interior-point solvers for semidefinite programming.     

Initialization in semidefinite programming via a self-dual skew-symmetric embedding

The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for large-scale linear programming. Many interior point methods for linear programming have now been extended to the more general semidefinite case, but the initialization problem remained unsolved.In this paper we show that the initialization strategy of embedding the problem in a self-dual skew-symmetric problem can also be extended to the semidefinite case. This method also provides a solution for the initialization of quadratic programs and it is applicable to more general convex problems with conic formulation.     

Local convergence of predictor—corrector infeasible-interior-point algorithms for SDPs and SDLCPs

An example of an SDP (semidefinite program) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the Mizuno—Todd—Ye type predictor—corrector primal-dual interior-point method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A Mizuno—Todd—Ye type predictor—corrector infeasible-interior-point algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

The curvature integral and the complexity of linear complementarity problems

In this paper, we propose a predictor—corrector-type algorithm for solving the linear complementarity problem (LCP), and prove that the actual number of iterations needed by the algorithm is bounded from above and from below by a curvature integral along the central trajectory of the problem. This curvature integral is not greater than, and possibly smaller than, the best upper bound obtained in the literature to date.Corresponding author.This author's research was partially supported by Research Grant No. RP920068, National University of Singapore, Singapore.     

On the finite termination of the Douglas-Rachford method for the convex feasibility problem

In this paper we apply the Douglas–Rachford (DR) method to solve the problem of finding a point in the intersection of the interior of a closed convex cone and a closed convex set in an infinite-dimensional Hilbert space. For this purpose, we propose two variants of the DR method which can find a point in the intersection in a finite number of iterations. In order to analyse the finite termination of the methods, we use some properties of the metric projection and a result regarding the rate of convergence of fixed point iterations. As applications of the results, we propose the methods for solving the conic and semidefinite feasibility problems, which terminate at a solution in a finite number of iterations.     

Return-mapping algorithms for associative isotropic hardening plasticity using conic optimization

We present a mathematical programming approach for elastoplastic constitutive initial value problems. Consideration of the associative plasticity and a linear isotropic hardening model allowed us to formulate the local discrete constitutive equations as conic programs. Specifically, we demonstrate that implicit return-mapping schemes for well-known yield criteria, such as the Rankine, von Mises, Tresca, Drucker-Prager, and Mohr–Coulomb criteria, can be expressed as second-order and semidefinite conic programs. Additionally, we propose a novel scheme for the numerical evaluation of the consistent elastoplastic tangent operator based on a first-order parameter derivative of the optimal solutions.     

Unifying Optimal Partition Approach to Sensitivity Analysis in Conic Optimization

We study convex conic optimization problems in which the right-hand side and the cost vectors vary linearly as functions of a scalar parameter. We present a unifying geometric framework that subsumes the concept of the optimal partition in linear programming (LP) and semidefinite programming (SDP) and extends it to conic optimization. Similar to the optimal partition approach to sensitivity analysis in LP and SDP, the range of perturbations for which the optimal partition remains constant can be computed by solving two conic optimization problems. Under a weaker notion of nondegeneracy, this range is simply given by a minimum ratio test. We discuss briefly the properties of the optimal value function under such perturbations.     

On two homogeneous self-dual approaches to linear programming and its extensions

We investigate the relation between interior-point algorithms applied to two homogeneous self-dual approaches to linear programming, one of which was proposed by Ye, Todd, and Mizuno and the other by Nesterov, Todd, and Ye. We obtain only a partial equivalence of path-following methods (the centering parameter for the first approach needs to be equal to zero or larger than one half), whereas complete equivalence of potential-reduction methods can be shown. The results extend to self-scaled conic programming and to semidefinite programming using the usual search directions. Received: July 1998 / Accepted: September 2000?Published online November 17, 2000     

Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems

We study four measures of problem instance behavior that might account for the observed differences in interior-point method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar-) condition measure C(d) of the data instance, (iii) a measure of the near-absence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the sample correlation (CORR) between these measures and IPM iteration counts (solved using the software SDPT3) when these measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.901), and provides a very good explanation of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The near-absence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations. This research has been partially supported through the MIT-Singapore Alliance.     

SOS-Convex Semialgebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization

In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with ? ₁-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex semialgebraic programs can be found by solving a single semidefinite programming problem (SDP). We achieve the results by using tools from semialgebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials. As an application, we show that robust SOS-convex optimization proble ms under restricted spectrahedron data uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP relaxation result for restricted ellipsoidal data uncertainty and answers an open question in the literature on how to recover a robust solution of uncertain SOS-convex polynomial programs from its semidefinite programming relaxation in this broader setting.     

Implementation of a block-decomposition algorithm for solving large-scale conic semidefinite programming problems

In this paper, we consider block-decomposition first-order methods for solving large-scale conic semidefinite programming problems given in standard form. Several ingredients are introduced to speed-up the method in its pure form such as: an aggressive choice of stepsize for performing the extragradient step; use of scaled inner products; dynamic update of the scaled inner product for properly balancing the primal and dual relative residuals; and proper choices of the initial primal and dual iterates, as well as the initial parameter for the scaled inner product. Finally, we present computational results showing that our method outperforms the two most competitive codes for large-scale conic semidefinite programs, namely: the boundary-point method introduced by Povh et al. and the Newton-CG augmented Lagrangian method by Zhao et al.     

一个求解半正定规划问题的新原始一对偶内点算法

在原始对偶内点算法的设计和分析中,障碍函数对算法的搜索方法和复杂性起着重要的作用。本文由核函数来确定障碍函数,设计了一个求解半正定规划问题的原始。对偶内点算法。这个障碍函数即可以定义算法新的搜索方向,又度量迭代点与中心路径的距离,同时对算法的复杂性分析起着关键的作用。我们计算了算法的迭代界,得出了关于大步校正法和小步校正法的迭代界,它们分别是O（√n log n log n/c）和O（√n log n/ε）,这里n是半正定规划问题的维数。最后,我们根据一个算例,说明了算法的有效性以及对核函数的参数的敏感性。     

Strong calmness of perturbed KKT system for a class of conic programming with degenerate solutions

ABSTRACT

This paper is concerned with the strong calmness of the KKT solution mapping for a class of canonically perturbed conic programming, which plays a central role in achieving fast convergence under situations when the Lagrange multiplier associated to a solution of these conic optimization problems is not unique. We show that the strong calmness of the KKT solution mapping is equivalent to a local error bound for the solutions to the perturbed KKT system, and is also equivalent to the pseudo-isolated calmness of the stationary point mapping along with the calmness of the multiplier set mapping at the corresponding reference point. Sufficient conditions are also provided for the strong calmness by establishing the pseudo-isolated calmness of the stationary point mapping in terms of the noncriticality of the associated multiplier, and the calmness of the multiplier set mapping in terms of a relative interior condition for the multiplier set. These results cover and extend the existing ones in Hager and Gowda [Stability in the presence of degeneracy and error estimation. Math Program. 1999;85:181–192]; Izmailov and Solodov [Stabilized SQP revisited. Math Program. 2012;133:93–120] for nonlinear programming and in Cui et al. [On the asymptotic superlinear convergence of the augmented Lagrangian method for semidefinite programming with multiple solutions. 2016, arXiv: 1610.00875v1]; Zhang and Zhang [Critical multipliers in semidefinite programming. 2018, arXiv: 1801.02218v1] for semidefinite programming.     

Sensitivity analysis in linear programming and semidefinite programming using interior-point methods

We analyze perturbations of the right-hand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the perturbations that allow interior-point methods to recover feasible and near-optimal solutions in a single interior-point iteration. For the unique, nondegenerate solution case in LP, we show that the bounds obtained using interior-point methods compare nicely with the bounds arising from using the optimal basis. We also present explicit bounds for SDP using the Monteiro-Zhang family of search directions and specialize them to the AHO, H..K..M, and NT directions. Received: December 1999 / Accepted: January 2001?Published online March 22, 2001