Optimizing condition numbers

In this paper we study the problem of minimizing condition numbers over a compact convex subset of the cone of symmetric positive semidefinite $n\times n$ matrices. We show that the condition number is a Clarke regular strongly pseudoconvex function. We prove that a global solution of the problem can be approximated by an exact or an inexact solution of a nonsmooth convex program. This asymptotic analysis provides a valuable tool for designing an implementable algorithm for solving the problem of minimizing condition numbers

Optimality conditions and constraint qualifications for cardinality constrained optimization problems

The cardinality constrained optimization problem (CCOP) is an optimization problem where the maximum number of nonzero components of any feasible point is bounded. In this paper, we consider CCOP as a mathematical program with disjunctive subspaces constraints (MPDSC). Since a subspace is a special case of a convex polyhedral set, MPDSC is a special case of the mathematical program with disjunctive constraints (MPDC). Using the special structure of subspaces, we are able to obtain more precise formulas for the tangent and (directional) normal cones for the disjunctive set of subspaces. We then obtain first and second order optimality conditions by using the corresponding results from MPDC. Thanks to the special structure of the subspace, we are able to obtain some results for MPDSC that do not hold in general for MPDC. In particular we show that the relaxed constant positive linear dependence (RCPLD) is a sufficient condition for the metric subregularity/error bound property for MPDSC which is not true for MPDC in general. Finally we show that under all constraint qualifications presented in this paper, certain exact penalization holds for CCOP

Directional subdifferential of the value function

The directional subdifferential of the value function gives an estimate on how much the optimal value changes under a perturbation in a certain direction. In this paper we derive upper estimates for the directional limiting and singular subdifferential of the value function for a very general parametric optimization problem. We obtain a characterization for the directional Lipschitzness of a locally lower semicontinuous function in terms of the directional subdifferentials. Based on this characterization and the derived upper estimate for the directional singular subdifferential, we are able to obtain a sufficient condition for the directional Lipschitzness of the value function. Finally, we specify these results for various cases when all functions involved are smooth, when the perturbation is additive, when the constraint is independent of the parameter, or when the constraints are equalities and inequalities. Our results extend the corresponding results on the sensitivity of the value function to allow directional perturbations. Even in the case of full perturbations, our results recover or even extend some existing results, including the Danskin's theorem

Spinal cord stimulation for predominant low back pain in failed back surgery syndrome: study protocol for an international multicenter randomized controlled trial (PROMISE study)

Background: Although results of case series support the use of spinal cord stimulation in failed back surgery syndrome patients with predominant low back pain, no confirmatory randomized controlled trial has been undertaken in this patient group to date. PROMISE is a multicenter, prospective, randomized, open-label, parallel-group study designed to compare the clinical effectiveness of spinal cord stimulation plus optimal medical management with optimal medical management alone in patients with failed back surgery syndrome and predominant low back pain. Method/Design: Patients will be recruited in approximately 30 centers across Canada, Europe, and the United States. Eligible patients with low back pain exceeding leg pain and an average Numeric Pain Rating Scale score >= 5 for low back pain will be randomized 1:1 to spinal cord stimulation plus optimal medical management or to optimal medical management alone. The investigators will tailor individual optimal medical management treatment plans to their patients. Excluded from study treatments are intrathecal drug delivery, peripheral nerve stimulation, back surgery related to the original back pain complaint, and experimental therapies. Patients randomized to the spinal cord stimulation group will undergo trial stimulation, and if they achieve adequate low back pain relief a neurostimulation system using the Specify (R) 5-6-5 multi-column lead (Medtronic Inc., Minneapolis, MN, USA) will be implanted to capture low back pain preferentially in these patients. Outcome assessment will occur at baseline (pre-randomization) and at 1, 3, 6, 9, 12, 18, and 24 months post randomization. After the 6-month visit, patients can change treatment to that received by the other randomized group. The primary outcome is the proportion of patients with >= 50% reduction in low back pain at the 6-month visit. Additional outcomes include changes in low back and leg pain, functional disability, health-related quality of life, return to work, healthcare utilization including medication usage, and patient satisfaction. Data on adverse events will be collected. The primary analysis will follow the intention-to-treat principle. Healthcare use data will be used to assess costs and long-term cost-effectiveness. Discussion: Recruitment began in January 2013 and will continue until 2016