convex extensions and envelopes of lower semi continuous functions we define a convex extension of a lower semi continuous function to be a convex function that is identical to the given function over a pre specified subset of its domain convex extensions are not necessarily constructible or unique we identify conditions under which a convex extension can be constructed when multiple convex extensions exist we characterize the tightest convex extension in a well defined sense using the notion of a generating set we establish conditions under which the tightest convex extension is the convex envelope then we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions finally using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of x y over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outer linearization of bilinear terms

lp parameter perturbation and design of robust controllers for linear systems in this paper linear systems with characteristic equations that are linearly dependent on some uncertain parameters are considered a new procedure is proposed for the calculation of d stability margins in the parameter space in general lp norms the procedure is simple and computationally efficient for several important regions of interest in the complex plane e g left hand plane sectors and circles centred at the origin it requires the solution of only one equation based on this result a new method is presented for the design of robust controllers robust placement of the closed loop poles inside the regions of interest is addressed the method provides a simple and computationally tractable solution to the robust controller design problem it can be used to synthesize robust controllers with general rational transfer functions for plants modelled as the ratio of two uncertain polynomials subjected to lp parameter perturbation

damping margins of interval polynomials the zero locations of interval polynomials are examined in particular it is shown that a family of interval polynomials will have zeros only in the left sector if the real and imaginary parts of four specially constructed complex polynomials have an interlacing real zero property this is significant for the analysis of uncertain systems as the computation cost associated with checking the zero locations of interval polynomials will be greatly reduced the results presented can be readily extended to more general stability regions where the real and imaginary parts of the polar plot are polynomial functions

a generalization of strong kharitonov theorems to polytopes of polynomials the root clustering property of a certain class of polytopes of polynomials is examined it is shown that the polytopes of polynomials have only zeros within the left sector if and only if a subset of their edge polynomials has only zeros within the left sector certain conditions are derived under which the root clustering property of the polytopes of polynomials can be inferred from a subset of their vertices

a convex analysis perspective on disjunctive cuts we treat with tools from convex analysis the general problem of cutting planes separating a point from a closed convex set p crucial for this is the computation of extreme points in the so called reverse polar set introduced by e balas in 1979 in the polyhedral case this enables the computation of cuts that define facets of p we exhibit three equivalent optimization problems to compute such extreme points one of them corresponds to selecting a specific normalization to generate cuts we apply the above development to the case where p is the closed convex hull of a union and more particularly a union of polyhedra case of disjunctive cuts we conclude with some considerations on the design of efficient cut generators the paper also contains an appendix reviewing some fundamental concepts of convex analysis

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

geometrical and spectral study of beta skeleton graphs we perform an extensive numerical analysis of skeleton graphs a particular type of proximity graphs in a skeleton graph bsg two vertices are connected if a proximity rule that depends of the parameter 0 is satisfied moreover for 1 there exist two different proximity rules leading to lune based and circle based bsgs first by computing the average degree of large ensembles of bsgs we detect differences which increase with the increase of between lune based and circle based bsgs then within a random matrix theory rmt approach we explore spectral and eigenvector properties of random bsgs by the use of the nearest neighbor energy level spacing distribution and the entropic eigenvector localization length respectively the rmt analysis allows us to conclude that a localization transition occurs at 1

root clustering of interval polynomials in the left sector abstract we examine the root clustering property of interval polynomials in particular we shall show that a family of interval polynomials will have only zeros within the left sector if and only if a finite number of specially chosen vertex polynomials have only zeros in the left sector the number of required vertex polynomials is a linear function of the orders of the polynomial but is independent of the damping margins of the left sector

strong valid inequalities for orthogonal disjunctions and bilinear covering sets in this paper we derive a closed form characterization of the convex hull of a generic nonlinear set when this convex hull is completely determined by orthogonal restrictions of the original set although the tools used in this construction include disjunctive programming and convex extensions our characterization does not introduce additional variables we develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed we demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed integer polyhedral sets and finding the convex hull of certain mixed pure integer bilinear sets we then extend the utility of the convexification tool to relaxing nonconvex inequalities which are not naturally disjunctive by providing sufficient conditions for establishing the convex extension property over the non negative orthant we illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non negative orthant although we illustrate our results primarily on bilinear covering sets they also apply to more general polynomial covering sets for which they yield new tight relaxations

a polyhedral branch and cut approach to global optimization a variety of nonlinear including semidefinite relaxations have been developed in recent years for nonconvex optimization problems their potential can be realized only if they can be solved with sufficient speed and reliability unfortunately state of the art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software in this paper we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch and cut approach our algorithm exploits convexity either identified automatically or supplied through a suitable modeling language construct in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems we prove that if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function the convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited root node relaxations are computed for 87 problems from globallib and minlplib and detailed computational results are presented for globally solving 26 of these problems with baron 7 2 which implements the proposed techniques the use of cutting planes for these problems reduces root node relaxation gaps by up to 100 and expedites the solution process often by several orders of magnitude

a precise correspondence between lift and project cuts simple disjunctive cuts and mixed integer gomory cuts for 0 1 programming we establish a precise correspondence between lift and project cuts for mixed 0 1 programs simple disjunctive cuts intersection cuts and mixed integer gomory cuts the correspondence maps members of one family onto members of the others it also maps bases of the higher dimensional cut generating linear program cglp into bases of the linear programming relaxation it provides new bounds on the number of facets of the elementary closure and on the rank of the standard linear programming relaxation of the mixed 0 1 polyhedron with respect to the above families of cutting planes

on unions and dominants of polytopes a well known result on unions of polyhedra in the same space gives an extended formulation in a higher dimensional space whose projection is the convex hull of the union here in contrast we study the unions of polytopes in different spaces giving a complete description of the convex hull without additional variables when the underlying polytopes are monotone the facets are described explicitly generalizing results of hong and hooker on cardinality rules and an efficient separation algorithm is proposed these results are based on an explicit representation of the dominant of a polytope and an algorithm for the separation problem for the dominant for non monotone polytopes both the dominant and the union are characterized we also give results on the unions of polymatroids both on disjoint ground sets and on the same ground set generalizing results of conforti and laurent

necessary and sufficient conditions for root clustering of a polytope of polynomials in a simply connected domain linear time invariant systems with uncertain parameters are considered a test is given which is both necessary and sufficient for root clustering of a family of polytopic polynomials in a simply connected domain the test does not require checking a continuum of polynomials special cases include stability testing of continuous time systems and of discrete time systems with uncertain parameters as well as relative stability tests of such systems interval polynomials can also be handled as a special case

frequency domain criteria for l sup p robust stability of continuous linear systems necessary and sufficient criteria in the frequency domain for robust stability are given under the assumption that coefficients of a characteristic polynomial belong to a transformed l sup p ball three cases are considered in detail p infinity interval uncertainty p 2 ellipsoidal uncertainty and p 1 octahedral uncertainty it is shown that frequency domain stability criteria are efficient when one deals with robust stability problems the frequency domain approach considered can be extended to various problems of robust stability

strong formulations of robust mixed 0 1 programming we introduce strong formulations for robust mixed 0 1 programming with uncertain objective coefficients we focus on a polytopic uncertainty set described by a budget constraint for allowed uncertainty in the objective coefficients we show that for a robust 0 1 problem there is an tight linear programming formulation with size polynomial in the size of an tight linear programming formulation for the nominal 0 1 problem we give extensions to robust mixed 0 1 programming and present computational experiments with the proposed formulations

convex programming for disjunctive convex optimization given a finite number of closed convex sets whose algebraic representation is known we study the problem of finding the minimum of a convex function on the closure of the convex hull of the union of those sets we derive an algebraic characterization of the feasible region in a higher dimensional space and propose a solution procedure akin to the interior point approach for convex programming

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

a branch and cut method for 0 1 mixed convex programming we generalize the disjunctive approach of balas ceria and cornuejols 2 and devevlop a branch and cut method for solving 0 1 convex programming problems we show that cuts can be generated by solving a single convex program we show how to construct regions similar to those of sherali and adams 20 and lovasz and schrijver 12 for the convex case finally we give some preliminary computational results for our method

different transformations for solving non convex trim loss problems by minlp in the present paper trim loss problems often named the cutting stock problem connected to the paper industry are considered the problem is to cut out a set of product paper rolls from raw paper rolls such that the cost function including the trim loss as well as the costs for the over production is minimized the problem is non convex due to certain bilinear constraints the problem can however be transformed into linear or convex form the resulting transformed problems can thereafter be solved as mixed integer linear programming problems or convex mixed integer non linear programming problems the linear and convex formulations are attractive from a formal point of view since global optimal solutions to the originally non convex problem can be obtained however as the examples considered will show the numerical efficiency of the solutions from the different transformed formulations varies considerably an example based on a trim optimization problem encountered daily at a finnish paper converting mill is finally presented in order to demonstrate differences in the numerical solutions

a note on two methods related to stability robustness of polynomials in a sector relative stability two recent results on the stability robustness of uncertain systems polynomials are addressed one gives a vertex test for left sector relative stability where the number of the required vertices is four for a real case and eight for a complex case the other gives an elegant frequency domain graphical hodograph approach for left plane stability where the plotting of only one two for a complex case hodograph plus simple boundary conditions is both necessary and sufficient to obtain the maximum coefficient perturbation bounds it is shown that a similar graphical test exists for the left sector stability case

generalization of strong kharitonov theorems to the left sector the zero locations of interval polynomials are examined in particular it is shown that a family of interval polynomials will have only zeros in a certain class of left sector if and only if a finite number of specially chosen vertex polynomials have only zeros in the left sector this finite number of vertex polynomials is dependent on the damping margins of the left sector but is independent of the orders of the polynomials the exact number of vertex polynomials to be checked will depend on the damping margins but it will be independent of the orders of the polynomials these specially chosen vertex polynomials are independent of the order of the polynomials the importance of the result lies with the great reduction in computation cost associated with checking the zero locations of interval polynomials