Approximate optimality conditions and stopping criteria in canonical DC programming

In this paper, we study approximate optimality conditions for the Canonical DC (CDC) optimization problem and their relationships with stopping criteria for a large class of solution algorithms for the problem. In fact, global optimality conditions for CDC are very often restated in terms of a non-convex optimization problem, which has to be solved each time the optimality of a given tentative solution has to be checked. Since this is in principle a costly task, it makes sense to only solve the problem approximately, leading to an inexact stopping criteria and therefore to approximate optimality conditions. In this framework, it is important to study the relationships between the approximation in the stopping criteria and the quality of the solutions that the corresponding approximated optimality conditions may eventually accept as optimal, in order to ensure that a small tolerance in the stopping criteria does not lead to a disproportionally large approximation of the optimal value of the CDC problem. We develop conditions ensuring that this is the case; these turn out to be closely related with the well-known concept of regularity of a CDC problem, actually coinciding with the latter if the reverse-constraint set is a polyhedron

Outer Approximation Algorithms for DC Programs and Beyond

We consider the well-known Canonical DC (CDC) optimization problem, relying on an alternative equivalent formulation based on a polar characterization of the constraint, and a novel generalization of this problem, which we name Single Reverse Polar problem (SRP). We study the theoretical properties of the new class of (SRP) problems, and contrast them with those of (CDC)problems. We introduce of the concept of ``approximate oracle'' for the optimality conditions of (CDC) and (SRP), and make a thorough study of the impact of approximations in the optimality conditions onto the quality of the approximate optimal solutions, that is the feasible solutions which satisfy them. Afterwards, we develop very general hierarchies of convergence conditions, similar but not identical for (CDC) and (SRP), starting from very abstract ones and moving towards more readily implementable ones. Six and three different sets of conditions are proposed for (CDC) and (SRP), respectively. As a result, we propose very general algorithmic schemes, based on approximate oracles and the developed hierarchies, giving rise to many different implementable algorithms, which can be proven to generate an approximate optimal value in a finite number of steps, where the error can be managed and controlled. Among them, six different implementable algorithms for (CDC) problems, four of which are new and can't be reduced to the original cutting plane algorithm for (CDC) and its modifications; the connections of our results with the existing algorithms in the literature are outlined. Also, three cutting plane algorithms for solving (SRP) problems are proposed, which seem to be new and cannot be reduced to each other

Simplex Tableau based approximate projection in Karmarkar's algorithm

Ankara : Department of Industrial Engineering and Institute of Engineering and Sciences, Bilkent Univ., 1990.Thesis (Master's) -- Bilkent University, 1990.Includes bibliographical references leaves 31-33In this thesis, our main concern is to develop a new implementation of Karmarkar’s LP Algorithm and compare it with the standard version. In the implementation, the “Simplex Tableau” information is used in the basic step of the algorithm, the projection. Instead of constructing the whole projection matrix, some of the orthogonal feasible directions are obtained by using the Simplex Tableau and to give an idea of its effectiveness, this approximation scheme is compared with the standard implementation of Karmarkar’s Algorithm, by D. Gay. The Simplex Tableau is also used to calculate a basic feasible solution at any iteration with a very modest cost.Günalay, YavuzM.S

Outer Approximation Algorithms for Canonical DC Problems

The paper discusses a general framework for outer approximation type algorithms for the canonical DC optimization problem. The algorithms rely on a polar reformulation of the problem and exploit an approximated oracle in order to check global optimality. Consequently, approximate optimality conditions are introduced and bounds on the quality of the approximate global optimal solution are obtained. A thorough analysis of properties which guarantee convergence is carried out; two families of conditions are introduced which lead to design six implementable algorithms, whose convergence can be proved within a unified framework

Optimal low-rank approximations of Bayesian linear inverse problems

In the Bayesian approach to inverse problems, data are often informative, relative to the prior, only on a low-dimensional subspace of the parameter space. Significant computational savings can be achieved by using this subspace to characterize and approximate the posterior distribution of the parameters. We first investigate approximation of the posterior covariance matrix as a low-rank update of the prior covariance matrix. We prove optimality of a particular update, based on the leading eigendirections of the matrix pencil defined by the Hessian of the negative log-likelihood and the prior precision, for a broad class of loss functions. This class includes the F\"{o}rstner metric for symmetric positive definite matrices, as well as the Kullback-Leibler divergence and the Hellinger distance between the associated distributions. We also propose two fast approximations of the posterior mean and prove their optimality with respect to a weighted Bayes risk under squared-error loss. These approximations are deployed in an offline-online manner, where a more costly but data-independent offline calculation is followed by fast online evaluations. As a result, these approximations are particularly useful when repeated posterior mean evaluations are required for multiple data sets. We demonstrate our theoretical results with several numerical examples, including high-dimensional X-ray tomography and an inverse heat conduction problem. In both of these examples, the intrinsic low-dimensional structure of the inference problem can be exploited while producing results that are essentially indistinguishable from solutions computed in the full space

Optimal Control of Nonlinear Switched Systems: Computational Methods and Applications

A switched system is a dynamic system that operates by switching between different subsystems or modes. Such systems exhibit both continuous and discrete characteristics—a dual nature that makes designing effective control policies a challenging task. The purpose of this paper is to review some of the latest computational techniques for generating optimal control laws for switched systems with nonlinear dynamics and continuous inequality constraints. We discuss computational strategiesfor optimizing both the times at which a switched system switches from one mode to another (the so-called switching times) and the sequence in which a switched system operates its various possible modes (the so-called switching sequence). These strategies involve novel combinations of the control parameterization method, the timescaling transformation, and bilevel programming and binary relaxation techniques. We conclude the paper by discussing a number of switched system optimal control models arising in practical applications

On optimal designs for clinical trials: An updated review

Optimization of clinical trial designs can help investigators achieve higher qualityresults for the given resource constraints. The present paper gives an overviewof optimal designs for various important problems that arise in different stages ofclinical drug development, including phase I dose–toxicity studies; phase I/II studiesthat consider early efficacy and toxicity outcomes simultaneously; phase IIdose–response studies driven by multiple comparisons (MCP), modeling techniques(Mod), or their combination (MCP–Mod); phase III randomized controlled multiarmmulti-objective clinical trials to test difference among several treatment groups;and population pharmacokinetics–pharmacodynamics experiments. We find thatmodern literature is very rich with optimal design methodologies that can be utilizedby clinical researchers to improve efficiency of drug development