Compact relaxations for polynomial programming problems

Reduced RLT constraints are a special class of Reformulation- Linearization Technique (RLT) constraints. They apply to nonconvex (both continuous and mixed-integer) quadratic programming problems subject to systems of linear equality constraints. We present an extension to the general case of polynomial programming problems and discuss the derived convex relaxation. We then show how to perform rRLT constraint generation so as to reduce the number of inequality constraints in the relaxation, thereby making it more compact and faster to solve. We present some computational results validating our approach

On the composition of convex envelopes for quadrilinear terms

International audienceWithin the framework of the spatial Branch-and-Bound algorithm for solving Mixed-Integer Nonlinear Programs, different convex relaxations can be obtained for multilinear terms by applying associativity in different ways. The two groupings ((x1x2)x3)x4 and (x1x2x3)x4 of a quadrilinear term, for example, give rise to two different convex relaxations. In [6] we prove that having fewer groupings of longer terms yields tighter convex relaxations. In this paper we give an alternative proof of the same fact and perform a computational study to assess the impact of the tightened convex relaxation in a spatial Branch-and-Bound setting

Towards Machine Wald

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed \emph{by humans} because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to \emph{think} as \emph{humans} have the ability to do when faced with uncertainty is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables. (2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tend to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with Decision Theory, Machine Learning, Bayesian Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty Quantification and Information Based Complexity.Comment: 37 page

Error bounds for monomial convexification in polynomial optimization

Convex hulls of monomials have been widely studied in the literature, and monomial convexifications are implemented in global optimization software for relaxing polynomials. However, there has been no study of the error in the global optimum from such approaches. We give bounds on the worst-case error for convexifying a monomial over subsets of $[0,1]^n$. This implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately. Our main error bounds depend primarily on the degree of the monomial, making them easy to compute. Since monomial convexification studies depend on the bounds on the associated variables, in the second part, we conduct an error analysis for a multilinear monomial over two different types of box constraints. As part of this analysis, we also derive the convex hull of a multilinear monomial over $[-1,1]^n$.Comment: 33 pages, 2 figures, to appear in journa

On modeling incentive systems which utilize pollution charges for pollution abatement

This paper examines a scheme of economic incentives for environmental protection, in which spatially differentiated pollution taxes are in use in compensating the pollution abatement costs. A simple mathematical model is described which determines an incentive system that encourages polluters to reduce the discharges to an acceptable level of ambient quality standards in a “cost-effective” manner. It is shown that the vector of pollution charges has to be proportional to the marginal abatement cost vector, but is smaller than the latter in magnitude. It is demonstrated that a necessary incentive effect may be achieved even if the total pollution charge is much lower (about three times) than the total abatement costs. It is also estimated how this charge incentive system reconciles conflicting criteria of cost-effectiveness and of equity. These conclusions are verified by numerical experiments with real data. Copyright Kluwer Academic Publishers 1992Economic incentives, pollution charges, pollution control subsides, cost-effectiveness analysis,

MODERN TRENDS IN SURGICAL TREATMENT OF PATIENTS WITH ACL RUPTURES (LITERATURE REVIEW)

The authors conducted an analysis of national and foreign scientific publications dedicated to the problems in treatment of patients with ruptures of the anterior cruciate ligament of the knee joint. The results of the analysis demonstrated that such lesions still remain the key knee pathology resulting from sports injuries that significantly affect knee function and require timely reconstructive surgical correction. Based on the study the key areas of improvement in treatment for mentioned category of patients have been identified. This is the biomechanically justified single bundle anatomical ACL reconstruction which is currently widely applied in the clinical practice by using of an isometrically located autograft. Such technique represents a radically new stage in the development of treatment methods for young and middle-aged patients with high functional demands

RESULTS OF SHOULDER STABILIZATION BY A MODIFIED BRISTOW - LATARJET PROCEDURE WITH ARTHROSCOPY

The authors describe the minimally invasive technique for Bristow-Latarjet bone unfree autoplasty with arthroscopy in patients with bone loss more than 25% of anterior-posterior diameter of the glenoid, the poor quality of the capsule or deep defects of Hill-Sachs. The analysis of the early results of treatment in 19 patients and midterm results - in 13 soldiers operated in 2011-2014. Features of the proposed technique are the shortening of surgical approach and the reduction of subscapularis muscle damage. In addition, arthroscopic support allows to attain the precision location of the graft relative to the articular surface of scapula, at the same time restoring the damaged anatomy SLAP, rotator cuff tendons and posterior labrum and restore shoulder ligaments tension and isolate bone graft from the joint cavity, contributing to a better articulation of the humeral head and reducing the risk of nonunion and resorption