Gordon's inequality and condition numbers in conic optimization

The probabilistic analysis of condition numbers has traditionally been approached from different angles; one is based on Smale's program in complexity theory and features integral geometry, while the other is motivated by geometric functional analysis and makes use of the theory of Gaussian processes. In this note we explore connections between the two approaches in the context of the biconic homogeneous feasiblity problem and the condition numbers motivated by conic optimization theory. Key tools in the analysis are Slepian's and Gordon's comparision inequalities for Gaussian processes, interpreted as monotonicity properties of moment functionals, and their interplay with ideas from conic integral geometry

Variational approximation of functionals defined on 1-dimensional connected sets: the planar case

In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a $\Gamma$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to $n$-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure

Effective Condition Number Bounds for Convex Regularization

We derive bounds relating Renegar's condition number to quantities that govern the statistical performance of convex regularization in settings that include the $\ell_1$-analysis setting. Using results from conic integral geometry, we show that the bounds can be made to depend only on a random projection, or restriction, of the analysis operator to a lower dimensional space, and can still be effective if these operators are ill-conditioned. As an application, we get new bounds for the undersampling phase transition of composite convex regularizers. Key tools in the analysis are Slepian's inequality and the kinematic formula from integral geometry.Comment: 17 pages, 4 figures . arXiv admin note: text overlap with arXiv:1408.301

Computational Methods for Discrete Conic Optimization Problems

This thesis addresses computational aspects of discrete conic optimization. Westudy two well-known classes of optimization problems closely related to mixedinteger linear optimization problems. The case of mixed integer second-ordercone optimization problems (MISOCP) is a generalization in which therequirement that solutions be in the non-negative orthant is replaced by arequirement that they be in a second-order cone. Inverse MILP, on the otherhand, is the problem of determining the objective function that makes a givensolution to a given MILP optimal.Although these classes seem unrelated on the surface, the proposedsolution methodology for both classes involves outer approximation of a conicfeasible region by linear inequalities. In both cases, an iterative algorithmin which a separation problem is solved to generate the approximation isemployed. From a complexity standpoint, both MISOCP and inverse MILP areNP--hard. As in the case of MILPs, the usual decision version ofMISOCP is NP-complete, whereas in contrast to MILP, we provide the firstproof that a certain decision version of inverse MILP is rathercoNP-complete.With respect to MISOCP, we first introduce a basic outer approximationalgorithm to solve SOCPs based on a cutting-plane approach. As expected, theperformance of our implementation of such an algorithm is shown to lag behindthe well-known interior point method. Despite this, such a cutting-planeapproach does have promise as a method of producing bounds when embedded withina state-of-the-art branch-and-cut implementation due to its superior ability towarm-start the bound computation after imposing branching constraints. Ourouter-approximation-based branch-and-cut algorithm relaxes both integrality andconic constraints to obtain a linear relaxation. This linear relaxation isstrengthened by the addition of valid inequalities obtained by separatinginfeasible points. Valid inequalities may be obtained by separation from theconvex hull of integer solution lying within the relaxed feasible region or byseparation from the feasible region described by the (relaxed) conicconstraints. Solutions are stored when both integer and conic feasibility isachieved. We review the literature on cutting-plane procedures for MISOCP andmixed integer convex optimization problems.With respect to inverse MILP, we formulate this problem as a conicproblem and derive a cutting-plane algorithm for it. The separation problem inthis algorithm is a modified version of the original MILP. We show that thereis a close relationship between this algorithm and a similar iterativealgorithm for separating infeasible points from the convex hull of solutions tothe original MILP that forms part of the basis for the well-known result ofGrotschel-Lovasz-Schrijver that demonstrates the complexity-wiseequivalence of separation and optimization.In order to test our ideas, we implement a number of software librariesthat together constitute DisCO, a full-featured solver for MISOCP. Thefirst of the supporting libraries is OsiConic, an abstract base classin C++ for interfacing to SOCP solvers. We provide interfaces using thislibrary for widely used commercial and open source SOCP/nonlinear problemsolvers. We also introduce CglConic, a library that implements cuttingprocedures for MISOCP feasible set. We perform extensive computationalexperiments with DisCO comparing a wide range of variants of our proposedalgorithm, as well as other approaches. As DisCO is built on top of a libraryfor distributed parallel tree search algorithms, we also perform experimentsshowing that our algorithm is effective and scalable when parallelized

Variational Approximation of Functionals Defined on 1-dimensional Connected Sets: The Planar Case

In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert--Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica--Mortola type energies proving a $Gamma$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to $n$-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [M. Bonafini, G. Orlandi, and E. Oudet, Variational Approximation of Functionals Defined on 1-Dimensional Connected Sets in $mathbb{R}^n$, preprint, 2018]

Gaussian Phase Transitions and Conic Intrinsic Volumes: Steining the Steiner Formula

Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical inference. It is a well-known fact that, given a closed convex cone $C\subset \mathbb{R}^d$, its conic intrinsic volumes determine a probability measure on the finite set $\{0,1,...d\}$, customarily denoted by $\mathcal{L}(V_C)$. The aim of the present paper is to provide a Berry-Esseen bound for the normal approximation of ${\cal L}(V_C)$, implying a general quantitative central limit theorem (CLT) for sequences of (correctly normalised) discrete probability measures of the type $\mathcal{L}(V_{C_n})$, $n\geq 1$. This bound shows that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution. Our approach is based on a variety of techniques, namely: (1) Steiner formulae for closed convex cones, (2) Stein's method and second order Poincar\'e inequality, (3) concentration estimates, and (4) Fourier analysis. Our results explicitly connect the sharp phase transitions, observed in many regularised linear inverse problems with convex constraints, with the asymptotic Gaussian fluctuations of the intrinsic volumes of the associated descent cones. In particular, our findings complete and further illuminate the recent breakthrough discoveries by Amelunxen, Lotz, McCoy and Tropp (2014) and McCoy and Tropp (2014) about the concentration of conic intrinsic volumes and its connection with threshold phenomena. As an additional outgrowth of our work we develop total variation bounds for normal approximations of the lengths of projections of Gaussian vectors on closed convex sets.Comment: 42 page