solving problems with semidefinite and related constraints using interior point methods for nonlinear programming in this paper we describe how to reformulate a problem that has second order cone and or semidefiniteness constraints in order to solve it using a general purpose interior point algorithm for nonlinear programming the resulting problems are smooth and convex and numerical results from the dimacs implementation challenge problems and sdplib are provided

a strong dual for conic mixed integer programs mixed integer conic programming is a generalization of mixed integer linear programming in this paper we present an extension of the duality theory for mixed integer linear programming see m guzelsoy and t k ralphs int j oper res taichung 4 2007 pp 118 137 g l nemhauser and l a wolsey integer and combinatorial optimization wiley interscience new york 1988 to the case of mixed integer conic programming in particular we construct a subadditive dual for mixed integer conic programming problems under a simple condition on the primal problem we show that strong duality holds

dimension reduction and coefficient estimation in multivariate linear regression summary we introduce a general formulation for dimension reduction and coefficient estimation in the multivariate linear model we argue that many of the existing methods that are commonly used in practice can be formulated in this framework and have various restrictions we continue to propose a new method that is more flexible and more generally applicable the method proposed can be formulated as a novel penalized least squares estimate the penalty that we employ is the coefficient matrix s ky fan norm such a penalty encourages the sparsity among singular values and at the same time gives shrinkage coefficient estimates and thus conducts dimension reduction and coefficient estimation simultaneously in the multivariate linear model we also propose a generalized cross validation type of criterion for the selection of the tuning parameter in the penalized least squares simulations and an application in financial econometrics demonstrate competitive performance of the new method an extension to the non parametric factor model is also discussed

on mathematical programming with indicator constraints in this paper we review the relevant literature on mathematical optimization with logical implications i e where constraints can be either active or disabled depending on logical conditions to hold in the case of convex functions the theory of disjunctive programming allows one to formulate these logical implications as convex nonlinear programming problems in a space of variables lifted with respect to its original dimension we concentrate on the attempt of avoiding the issue of dealing with large nlps in particular we review some existing results that allow to work in the original space of variables for two relevant special cases where the disjunctions corresponding to the logical implications have two terms then we significantly extend these special cases in two different directions one involving more general convex sets and the other with disjunctions involving three terms computational experiments comparing disjunctive programming formulations in the original space of variables with straightforward bigm ones show that the former are computationally viable and promising

a block approach for single machine scheduling with release dates and due dates abstract the problem of scheduling jobs on a single machine subject to given release dates and due dates is considered a new algorithm of the branch and bound type for solving this problem is proposed the approach used in the paper is fully based on the eliminative properties of a block of jobs and in contradiction to the existing algorithms it does not require any heuristic method to generate a current solution in a search tree however a heuristic may be used in order to obtain a good initial solution extensive computational experiments show a high efficiency of the algorithm

sufficiency of cut generating functions the concept of cut generating function has its origin in the work of gomory and johnson from the 1970s it has received renewed attention in the past few years recently conforti cornuejols daniilidis lemarechal and malick proposed a general framework for studying cut generating functions however they gave an example showing that not all cuts can be produced by cut generating functions in this framework they conjectured a natural condition under which cut generating functions might be sufficient this note settles this open problem

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

on sublinear inequalities for mixed integer conic programs this paper studies mathcal k k sublinear inequalities a class of inequalities with strong relations to mathcal k k minimal inequalities for disjunctive conic sets we establish a stronger result on the sufficiency of mathcal k k sublinear inequalities that is we show that when mathcal k k is the nonnegative orthant or the second order cone mathcal k k sublinear inequalities together with the original conic constraint are always sufficient for the closed convex hull description of the associated disjunctive conic set when mathcal k k is the nonnegative orthant mathcal k k sublinear inequalities are tightly connected to functions that generate cuts so called cut generating functions in particular we introduce the concept of relaxed cut generating functions and show that each mathbb r n r n sublinear inequality is generated by one of these we then relate the relaxed cut generating functions to the usual ones studied in the literature recently under a structural assumption cornuejols wolsey and yildiz established the sufficiency of cut generating functions in terms of generating all nontrivial valid inequalities of disjunctive sets where the underlying cone is nonnegative orthant we provide an alternate and straightforward proof of this result under the same assumption as a consequence of the sufficiency of mathbb r n r n sublinear inequalities and their connection with relaxed cut generating functions

computational experience with an interior point cutting plane algorithm there has been a great deal of success in the last 20 years with the use of cutting plane algorithms to solve specialized integer programming problems generally these algorithms work by solving a sequence of linear programming relaxations of the integer programming problem and they use the simplex algorithm to solve the relaxations in this paper we describe experiments using a predictor corrector interior point method to solve the relaxations for some problems the interior point code requires considerably less time than a simplex based cutting plane algorithm

valid inequalities for mixed integer linear programs this tutorial presents a theory of valid inequalities for mixed integer linear sets it introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift and project cuts gomory mixed integer cuts mixed integer rounding cuts split cuts and intersection cuts and it reveals the relationships between these families the tutorial also discusses computational aspects of generating the cuts and their strength

maximum relevance minimum common redundancy feature selection for nonlinear data abstract in recent years feature selection based on relevance redundancy trade off criteria has become a very promising and popular approach in the field of machine learning however the existing algorithmic frameworks of mutual information feature selection have certain limitations for the common feature selection problems in practice to overcome these limitations the idea of a new framework is developed by introducing a novel maximum relevance and minimum common redundancy criterion and a minimax nonlinear optimization approach in particular a novel mutual information feature selection method based on the normalization of the maximum relevance and minimum common redundancy n mrmcr mi is presented which produces a normalized value in the range 0 1 and results in a regression problem we perform extensive experimental comparisons over numerous state of art algorithms using different forecasts bayesian additive regression tree treed gaussian process k nn and svm and different data sets two simulated and five real datasets the results show that the proposed algorithm outperforms the others in terms of feature selection and forecasting accuracy

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

semidefinite programming in combinatorial optimization we discuss the use of semidefinite programming for combinatorial optimization problems the main topics covered include i the lovasz theta function and its applications to stable sets perfect graphs and coding theory ii the automatic generation of strong valid inequalities iii the maximum cut problem and related problems and iv the embedding of finite metric spaces and its relationship to the sparsest cut problem

convex relaxations of non convex mixed integer quadratically constrained programs extended formulations this paper addresses the problem of generating strong convex relaxations of mixed integer quadratically constrained programming miqcp problems miqcp problems are very difficult because they combine two kinds of non convexities integer variables and non convex quadratic constraints to produce strong relaxations of miqcp problems we use techniques from disjunctive programming and the lift and project methodology in particular we propose new methods for generating valid inequalities from the equation y x x t we use the non convex constraint y x x t preccurlyeq 0 to derive disjunctions of two types the first ones are directly derived from the eigenvectors of the matrix y x x t with positive eigenvalues the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction we also use the convex sdp constraint y x x t succcurlyeq 0 to derive convex quadratic cuts and we combine both approaches in a cutting plane algorithm we present computational results to illustrate our findings

cardinality constrained linear quadratic optimal control as control implementation often incurs not only a variable cost associated with the magnitude or energy of the control but also a setup cost we consider a discrete time linear quadratic lq optimal control problem with a limited number of control implementations termed in this technical note the cardinality constrained linear quadratic optimal control cclq we first derive a semi analytical feedback policy for cclq problems using dynamic programming dp due to the exponential growth of the complexity in calculating the action regions however dp procedure is only efficient for cclq problems with a scalar state space recognizing this fact we develop then two lower bounding schemes and integrate them into a branch and bound bnb solution framework to offer an efficient algorithm in solving general cclq problems adopting the devised solution algorithm for cclq problems we can solve efficiently the linear quadratic optimal control problem with setup costs

strong formulations for quadratic optimization with m matrices and indicator variables we study quadratic optimization with indicator variables and an m matrix i e a psd matrix with non positive off diagonal entries which arises directly in image segmentation and portfolio optimization with transaction costs as well as a substructure of general quadratic optimization problems we prove under mild assumptions that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem to strengthen the formulation we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each and provide the convex hull descriptions of these sets we also describe strong conic quadratic valid inequalities preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation

cutting planes for optimization of convex functions over nonconvex sets we derive linear inequality characterizations for sets of the form mathop rm conv x q in mathbb r d times mathbb r q ge q x x in mathbb r d mathop rm int p where q is convex and differentiable and p subset mathbb r d we show that in several cases our characterization leads to polynomial time separation algorithms that operate in the original space of variables in particular when q is a positive definite quadratic and p is a polyhedron or an ellipsoid

two term disjunctions on the second order cone balas introduced disjunctive cuts in the 1970s for mixed integer linear programs several recent papers have attempted to extend this work to mixed integer conic programs in this paper we study the structure of the convex hull of a two term disjunction applied to the second order cone and develop a methodology to derive closed form expressions for convex inequalities describing the resulting convex hull our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex possibly nonlinear valid inequalities that correspond to these linear inequalities we identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull in particular our results on two term disjunctions on the second order cone generalize related results on split cuts by modaresi kilinc and vielma and by andersen and jensen

support vector machines with the ramp loss and the hard margin loss in the interest of deriving classifiers that are robust to outlier observations we present integer programming formulations of vapnik s support vector machine svm with the ramp loss and hard margin loss the ramp loss allows a maximum error of 2 for each training observation while the hard margin loss calculates error by counting the number of training observations that are in the margin or misclassified outside of the margin svm with these loss functions is shown to be a consistent estimator when used with certain kernel functions in computational studies with simulated and real world data svm with the robust loss functions ignores outlier observations effectively providing an advantage over svm with the traditional hinge loss when using the linear kernel despite the fact that training svm with the robust loss functions requires the solution of a quadratic mixed integer program qmip and is np hard while traditional svm requires only the solution of a continuous quadratic program qp we are able to find good solutions and prove optimality for instances with up to 500 observations solution methods are presented for the new formulations that improve computational performance over industry standard integer programming solvers alone

solving quadratic 0 1 problems by semidefinite programs and cutting planes we present computational experiments for solving quadratic 0 1 problems our approach combines a semidefinite relaxation with a cutting plane technique and is applied in a branch and bound setting our experiments indicate that this type of approach is very robust and allows to solve many moderately sized problems having say less than 100 binary variables in a routine manner 1998 the mathematical programming society inc published by elsevier science b v

solving nonlinear single unit commitment problems with ramping constraints we present a dynamic programming algorithm for solving the single unit commitment 1uc problem with ramping constraints and arbitrary convex cost functions the algorithm is based on a new approach for efficiently solving the single unit economic dispatch ed problem with ramping constraints and arbitrary convex cost functions improving on previously known ones that were limited to piecewise linear functions for simple convex functions such as the quadratic ones typically used in applications the solution cost of all the involved ed problems consisting of finding an optimal

robust classification and regression using support vector machines abstract in this paper we investigate the theoretical aspects of robust classification and robust regression using support vector machines given training data x 1 y 1 x l y l where l represents the number of samples x i r n and y i 1 1 for classification or y i r for regression we investigate the training of a support vector machine in the case where bounded perturbation is added to the value of the input x i r n we consider both cases where our training data are either linearly separable and nonlinearly separable respectively we show that we can perform robust classification or regression by using linear or second order cone programming

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

convex hull of two quadratic or a conic quadratic and a quadratic inequality in this paper we consider an aggregation technique introduced by yildiran j math control inf 26 417 450 2009 to study the convex hull of regions defined by two quadratic inequalities or by a conic quadratic and a quadratic inequality yildiran 2009 shows how to characterize the convex hull of open sets defined by two strict quadratic inequalities using linear matrix inequalities we show how this aggregation technique can be easily extended to yield valid conic quadratic inequalities for the convex hull of open sets defined by two strict quadratic inequalities or by a strict conic quadratic and a strict quadratic inequality we also show that for sets defined by a strict conic quadratic and a strict quadratic inequality under one additional containment assumption these valid inequalities characterize the convex hull exactly we also show that under certain topological assumptions the results from the open setting can be extended to characterize the closed convex hull of sets defined with non strict conic and quadratic inequalities

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

how good is the goemans williamson max cut algorithm the celebrated semidefinite programming algorithm for max cut introduced by goemans and williamson was known to have a performance ratio of at least alpha frac 2 pi min 0 theta le pi frac theta 1 cos theta 0 87856 alpha 0 87857 the exact performance ratio was unknown we prove that the performance ratio of their algorithm is exactly alpha furthermore we show that it is impossible to add valid linear constraints to improve the performance ratio

on minimal valid inequalities for mixed integer conic programs we study disjunctive conic sets involving a general regular closed convex full dimensional and pointed cone x1d4a6 such as the nonnegative orthant the lorentz cone or the positive semidefinite cone in a unified framework we introduce x1d4a6 minimal inequalities and show that under mild assumptions these inequalities together with the trivial cone implied inequalities are sufficient to describe the convex hull we focus on the properties of x1d4a6 minimal inequalities by establishing algebraic necessary conditions for an inequality to be x1d4a6 minimal this characterization leads to a broader algebraically defined class of x1d4a6 sublinear inequalities we demonstrate a close connection between x1d4a6 sublinear inequalities and the support functions of convex sets with a particular structure this connection results in practical ways of verifying x1d4a6 sublinearity and or x1d4a6 minimality of inequalities our study generalizes some of the results from the mixed integer linear case it is well known that the minimal inequalities for

a fast tabu search algorithm for the permutation flow shop problem abstract a fast and easily implementable approximation algorithm for the problem of finding a minimum makespan in the permutation flow shop is presented the algorithm is based on a tabu search technique with a specific neighborhood definition which employs a block of jobs notion computational experiments up to 500 jobs and 20 machines show its excellent numerical properties

robust portfolio selection problems in this paper we show how to formulate and solve robust portfolio selection problems the objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters we introduce uncertainty structures for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second order cone programs and therefore the computational effort required to solve them is comparable to that required for solving convex quadratic programs moreover we show that these uncertainty structures correspond to confidence regions associated with the statistical procedures employed to estimate the market parameters finally we demonstrate a simple recipe for efficiently computing robust portfolios given raw market data and a desired level of confidence

cut generating functions for integer variables for an integer linear program gomory s corner relaxation is obtained by ignoring the nonnegativity of the basic variables in a tableau formulation in this paper we do not relax these nonnegativity constraints we generalize a classical result of gomory and johnson characterizing minimal cut generating functions in terms of subadditivity symmetry and periodicity our result is based on the notion of generalized symmetry condition we also prove a 2 slope theorem for extreme cut generating functions in our setting in the spirit of the 2 slope theorem of gomory and johnson

intersection cuts for nonlinear integer programming convexification techniques for structured sets we study the generalization of split k branch split and intersection cuts from mixed integer linear programming to the realm of mixed integer nonlinear programming constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure we introduce two techniques to give precise characterizations of such convex hulls and use them to construct split k branch split and intersection cuts for several classes of non polyhedral sets in particular we give simple formulas for split cuts for essentially all convex sets described by a single conic quadratic inequality we also give simple formulas for k branch split cuts and some general intersection cuts for a wide variety of convex quadratic sets

spatial properties of velocity structure functions in turbulent wake flows in this paper we present experimental evidence that the scaling laws for the velocity structure functions s n r v x r ensuremath v x n n 2 4 6 8 hold in various parts of the flow domain the exponents that characterize the scaling are however a function of the position in the wake that is the local strength and ubiquity of coherent structures this variation is shown to be systematic and considerably exceeds the inaccuracy involved in the determination of the exponents this is an objective indication of the influence that the organized flow structures and inhomogeneity may have on intermittency in the analysis we invoke the concept of the extended self similarity ess

regression model selection a residual likelihood approach summary we obtain the residual information criterion ric a selection criterion based on the residual log likelihood for regression models including classical regression models box cox transformation models weighted regression models and regression models with autoregressive moving average errors we show that ric is a consistent criterion and that simulation studies for each of the four models indicate that ric provides better model order choices than the akaike information criterion corrected akaike information criterion final prediction error cp and r 2 except when the sample size is small and the signal to noise ratio is weak in this case none of the criteria performs well monte carlo results also show that ric is superior to the consistent bayesian information criterion bic when the signal to noise ratio is not weak and it is comparable with bic when the signal to noise ratio is weak and the sample size is large

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

the one machine sequencing problem let m 1 and m 3 be non bottleneck machines and m 2 a bottleneck machine processing only one job at a time suppose that n jobs have to be processed on m 1 m 2 and m 1 in that order and job i has to spend a time a on m 1 d 1 on m 2 and q 1 on m 3 we want to minimize the makespan this problem is important since its resolution provides a bound on the makespan of complicated systems such as job shops it is np hard in the strong sense however efficient branch and bound methods exist and we describe one of them our bound for the tree search is very close to the bound used by florian et al but the principle of branching is quite different at every node we construct by an o n log n algorithm a schrage schedule then we define a critical job c a critical set j and consider two subsets of schedules the schedules when c procedes every job of j and the schedules when c follows every job of j we give the results of this method and prove that the difference between the optimum and the schrage schedule is less than d 1

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

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

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

quadratic convex reformulations for semicontinuous quadratic programming we consider in this paper a class of semicontinuous quadratic programming problems which arises in many real world applications such as production planning portfolio selection and subset selection in regression we build upon the idea of the quadratic convex reformulation approach i e adding to the original objective function an additional convex term that is zero at all points of the feasible region exploiting the structure of semicontinuous variables we derive the most general set of quadratic functions that can be added we also incorporate the state of the art perspective reformulation into our new framework within this framework the set of optimal reformulations with the tightest continuous relaxation can be found by solving a semidefinite programming problem we also explore the relationship among different reformulations within our framework and simplify the search for optimal ones in certain cases with the help of second order cone programming among the proposed set of optimal reformula

mixed integer second order cone programming formulations for variable selection in linear regression abstract this study concerns a method of selecting the best subset of explanatory variables in a multiple linear regression model goodness of fit measures for example adjusted r2 aic and bic are generally used to evaluate a subset regression model although variable selection with regard to these measures is usually performed with a stepwise regression method it does not always provide the best subset of explanatory variables in this paper we propose mixed integer second order cone programming formulations for selecting the best subset of variables with respect to adjusted r2 aic and bic computational experiments show that in terms of these measures the proposed formulations yield better solutions than those provided by common stepwise regression methods

lifting for conic mixed integer programming lifting is a procedure for deriving valid inequalities for mixed integer sets from valid inequalities for suitable restrictions of those sets lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branch and cut algorithms here we generalize the theory of lifting to conic integer programming i e integer programs with conic constraints we show how to derive conic valid inequalities for a conic integer program from conic inequalities valid for its lower dimensional restrictions in order to simplify the computations we also discuss sequence independent lifting for conic integer programs when the cones are restricted to nonnegative orthants conic lifting reduces to the lifting for linear integer programming as one may expect

using simulated annealing to optimize the feature selection problem in marketing applications the feature selection also specification problem is concerned with finding the most influential subset of predictors in predictive modeling from a much larger set of potential predictors that can contain hundreds of predictors the problem belongs to the realm of combinatorial optimization where the objective is to find the subset of variables that optimize the value of some goodness of fit function due to the dimensionality of the problem the feature selection problem belongs to the group of np hard problems most of the available predictors are noisy or redundant and add very little if any to the prediction power of the model using all the predictors in the model often results in strong over fitting and very poor predictions constructing a prediction model by checking out all possible subsets is impractical due to computational volume looking on the contribution of each predictor separately is not accurate because it ignores the inter correlations between predictors as a result no analytic solution is available for the feature selection problem requiring that one resorts to heuristics in this paper we employ the simulated annealing sa approach which is one of the leading stochastic search methods for specifying a large scale linear regression model the sa results are compared to the results of the more common stepwise regression swr approach for model specification the models are applied on realistic data sets in database marketing we also use simulated data sets to investigate what data characteristics make the swr approach equivalent to the supposedly more superior sa approach

a conic integer programming approach to stochastic joint location inventory problems we study several joint facility location and inventory management problems with stochastic retailer demand in particular we consider cases with uncapacitated facilities capacitated facilities correlated retailer demand stochastic lead times and multicommodities we show how to formulate these problems as conic quadratic mixed integer problems valid inequalities including extended polymatroid and extended cover cuts are added to strengthen the formulations and improve the computational results compared to the existing modeling and solution methods the new conic integer programming approach not only provides a more general modeling framework but also leads to fast solution times in general

cuts for mixed 0 1 conic programming in this we paper we study techniques for generating valid convex constraints for mixed 0 1 conic programs we show that many of the techniques developed for generating linear cuts for mixed 0 1 linear programs such as the gomory cuts the lift and project cuts and cuts from other hierarchies of tighter relaxations extend in a straightforward manner to mixed 0 1 conic programs we also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts gomory cuts for mixed 0 1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems e g we show that all the subtour elimination inequalities for the traveling salesman problem are rank 1 gomory cuts with respect to a single semidefinite constraint we also include results from our preliminary computational experiments with these cuts

on the facets of mixed integer programs with two integer variables and two constraints in this paper we consider an infinite relaxation of the mixed integer linear program with two integer variables nonnegative continuous variables and two equality constraints and we give a complete characterization of its facets we also derive an analogous characterization of the facets of the underlying finite integer program

jackson s rule for single machine scheduling making a good heuristic better we consider the scheduling problem in which jobs with release dates and delivery times are to be scheduled on one machine we present a 4 3 approximation algorithm for the problem with precedence constraints among the jobs and two polynomial approximation schemes for the problem without precedence constraints at the core of each of the algorithms presented is jackson s rule a simple but seemingly robust heuristic for the problem

the chvatal gomory closure of a strictly convex body in this paper we prove that the chvatal gomory closure of a set obtained as an intersection of a strictly convex body and a rational polyhedron is a polyhedron thus we generalize a result of schrijver schrijver a 1980 on cutting planes ann discrete math 9 291 296 which shows that the chvatal gomory closure of a rational polyhedron is a polyhedron

on the chvatal gomory closure of a compact convex set in this paper we show that the chvatal gomory closure of any compact convex set is a rational polytope this resolves an open question of schrijver ann discret math 9 291 296 1980 for irrational polytopes and generalizes the same result for the case of rational polytopes schrijver in ann discret math 9 291 296 1980 rational ellipsoids dey and vielma in ipco xiv lecture notes in computer science vol 6080 springer berlin pp 327 340 2010 and strictly convex bodies dadush et al in math oper res 36 227 239 2011

how to convexify the intersection of a second order cone and a nonconvex quadratic a recent series of papers has examined the extension of disjunctive programming techniques to mixed integer second order cone programming for example it has been shown by several authors using different techniques that the convex hull of the intersection of an ellipsoid mathcal e e and a split disjunction l x j x j u le 0 l xj xj u 0 with l u l

an effective branch and bound algorithm for convex quadratic integer programming we present a branch and bound algorithm for minimizing a convex quadratic objective function over integer variables subject to convex constraints in a given node of the enumeration tree corresponding to the fixing of a subset of the variables a lower bound is given by the continuous minimum of the restricted objective function we improve this bound by exploiting the integrality of the variables using suitably defined lattice free ellipsoids experiments show that our approach is very fast on both unconstrained problems and problems with box constraints the main reason is that all expensive calculations can be done in a preprocessing phase while a single node in the enumeration tree can be processed in linear time in the problem dimension

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

perspective reformulations of mixed integer nonlinear programs with indicator variables we study mixed integer nonlinear programs minlp s that are driven by a collection of indicator variables where each indicator variable controls a subset of the decision variables an indicator variable when it is turned off forces some of the decision variables to assume fixed values and when it is turned on forces them to belong to a convex set many practical minlps contain integer variables of this type we first study a mixed integer set defined by a single separable quadratic constraint and a collection of variable upper and lower bound constraints and a convex hull description of this set is derived we then extend this result to produce an explicit characterization of the convex hull of the union of a point and a bounded convex set defined by analytic functions further we show that for many classes of problems the convex hull can be expressed via conic quadratic constraints and thus relaxations can be solved via second order cone programming our work is closely related with the earlier work of ceria and soares math program 86 595 614 1999 as well as recent work by frangioni and gentile math program 106 225 236 2006 and akturk et al oper res lett 37 187 191 2009 finally we apply our results to develop tight formulations of mixed integer nonlinear programs in which the nonlinear functions are separable and convex and in which indicator variables play an important role in particular we present computational results for three applications quadratic facility location network design with congestion and portfolio optimization with buy in thresholds that show the power of the reformulation technique

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

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

decompositions of semidefinite matrices and the perspective reformulation of nonseparable quadratic programs we study the problem of decomposing the hessian matrix of a mixed integer convex quadratic program micqp into the sum of positive semidefinite 2 2 matrices solving this problem enables the use

certifiably optimal sparse inverse covariance estimation we consider the maximum likelihood estimation of sparse inverse covariance matrices we demonstrate that current heuristic approaches primarily encourage robustness instead of the desired sparsity we give a novel approach that solves the cardinality constrained likelihood problem to certifiable optimality the approach uses techniques from mixed integer optimization and convex optimization and provides a high quality solution with a guarantee on its suboptimality even if the algorithm is terminated early using a variety of synthetic and real datasets we demonstrate that our approach can solve problems where the dimension of the inverse covariance matrix is up to 1000 s we also demonstrate that our approach produces significantly sparser solutions than glasso and other popular learning procedures makes less false discoveries while still maintaining state of the art accuracy

global optimization of mixed integer nonlinear programs a theoretical and computational study this work addresses the development of an efficient solution strategy for obtaining global optima of continuous integer and mixed integer nonlinear programs towards this end we develop novel relaxation schemes range reduction tests and branching strategies which we incorporate into the prototypical branch and bound algorithm in the theoretical algorithmic part of the paper we begin by developing novel strategies for constructing linear relaxations of mixed integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties we then use lagrangian linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming this theory leads to new range reduction schemes including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch and bound tree finally we incorporate these relaxation and reduction strategies in a branch and bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems in the computational part of the paper we describe our implementation discussing wherever appropriate the use of suitable data structures and associated algorithms we present computational experience with benchmark separable concave quadratic programs fractional 0 1 programs and mixed integer nonlinear programs from applications in synthesis of chemical processes engineering design just in time manufacturing and molecular design

fast numerical simulations of 2d turbulence using a dynamic model for subfilter motions we present numerical simulations of 2d turbulent flow using a new model for the subfilter scales which are computed using a dynamic equation linking the subfilter scales with the resolved velocity this equation is not postulated but derived from the constitutive equations under the assumption that the non linear interactions of subfilter scales between themselves are small compared to their distortions by the resolved scales such an assumption results in a linear stochastic equation for the subfilter scales which can be numerically solved by a decomposition of the subfilter scales into localized wave packets the wave packets are randomly produced by the smallest of the resolved scales they are further transported by the resolved scale velocity and they have wavenumbers and amplitudes which evolve according to the resolved strain performance of our model is compared with direct numerical simulations of decaying and forced turbulence for the same resolution numerical simulations using our model allow for a significant reduction of the computational time of the order of 100 in the case we consider and allow to achieve of significantly larger reynolds number than the direct method

maximizing a class of utility functions over the vertices of a polytope given a polytope x a monotone concave univariate function g and two vectors c and d we study the discrete optimization problem of finding a vertex of x that maximizes the utility function c x g d x this problem has numerous applications in combinatorial optimization with a probabilistic objective including estimation of project duration with stochastic times in reliability models in multinomial logit models and in robust optimization we show that the problem is x1d4a9 x1d4ab hard for any strictly concave function g even for simple polytopes such as the uniform matroid assignment and path polytopes and propose a 1 2 approximation algorithm for it we discuss improvements for special cases where g is the square root log utility negative exponential utility and multinomial logit probability function in particular for the square root function the approximation ratio is 4 5 we also propose a 1 25 approximation algorithm for a class of minimization problems in which the maximization of the utility func

outer product free sets for polynomial optimization and oracle based cuts this paper introduces cutting planes that involve minimal structural assumptions enabling the generation of strong polyhedral relaxations for a broad class of problems we consider valid inequalities for the set s cap p where s is a closed set and p is a polyhedron given an oracle that provides the distance from a point to s we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron these cuts are generated from convex forbidden zones or s free sets derived from the oracle we also consider the special case of polynomial optimization accordingly we develop a theory of outer product free sets where s is the set of real symmetric matrices of the form xx t all maximal outer product free sets of full dimension are shown to be convex cones and we identify several families of such sets these families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently computational experiments demonstrate the promise of our approach

regressions by leaps and bounds this paper describes several algorithms for computing the residual sums of squares for all possible regressions with what appears to be a minimum of arithmetic less than six floating point operations per regression and shows how two of these algorithms can be combined to form a simple leap and bound technique for finding the best subsets without examining all possible subsets the result is a reduction of several orders of magnitude in the number of operations required to find the best subsets

second order cone programming approaches for handling missing and uncertain data we propose a novel second order cone programming formulation for designing robust classifiers which can handle uncertainty in observations similar formulations are also derived for designing regression functions which are robust to uncertainties in the regression setting the proposed formulations are independent of the underlying distribution requiring only the existence of second order moments these formulations are then specialized to the case of missing values in observations for both classification and regression problems experiments show that the proposed formulations outperform imputation

sequencing of jobs in some production system abstract we consider a real life problem of scheduling clients orders in the production of concrete blocks in a factory of building industry this problem can be modelled as a hybrid flow shop scheduling problem with mixed no wait no store constraints and mixed bottleneck non bottleneck machines the objective function is to minimize maximum completion time to solve the problem we propose an approximation algorithm based on the tabu search approach

cones of matrices and successive convex relaxations of nonconvex sets let f be a compact subset of the n dimensional euclidean space rn represented by finitely or infinitely many quadratic inequalities we propose two methods one based on successive semidefinite programming sdp relaxations and the other on successive linear programming lp relaxations each of our methods generates a sequence of compact convex subsets ck k 1 2 of rn such that a the convex hull of f subseteq c k 1 subseteq c k monotonicity b cap k 1 infty c k text the convex hull of f asymptotic convergence our methods are extensions of the corresponding lovasz schrijver lift and project procedures with the use of sdp or lp relaxation applied to general quadratic optimization problems qops with infinitely many quadratic inequality constraints utilizing descriptions of sets based on cones of matrices and their duals we establish the exact equivalence of the sdp relaxation and the semi infinite convex qop relaxation proposed originally by fujie and kojima using this equivalence we investigate some fundamental features of the two methods including a and b above

fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems abstract branch and cut methods for integer programming problems solve a sequence of linear programming problems traditionally these linear programming relaxations have been solved using the simplex method the reduced costs available at the optimal solution to a relaxation may make it possible to fix variables at zero or one if the solution to a relaxation is fractional additional constraints can be generated which cut off the solution to the relaxation but donot cut off any feasible integer points gomory cutting planes and other classes of cutting planes are generated from the final tableau in this paper we consider using an interior point method to solve the linear programming relaxations we show that it is still possible to generate gomory cuts and other cuts without having to recreate a tableau and we also show how variables can be fixed without using the optimal reduced costs the procedures we develop do not require that the current relaxation be solved to optimality this is useful for an interior point method because early termination of the current relaxation results in an improved starting point for the next relaxation

lifting inequalities a framework for generating strong cuts for nonlinear programs in this paper we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs the theory is geometric and provides insights into lifting based cut generation procedures yielding short proofs of earlier results in mixed integer programming using convex extensions we obtain conditions that allow for sequence independent lifting in nonlinear settings paving a way for efficient cut generation procedures for nonlinear programs this sequence independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general purpose strong cuts for integer programs we specialize our lifting results to derive facet defining inequalities for mixed integer bilinear knapsack sets finally we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut generation techniques applied on a tight reformulation of the problem

using intelligent backtracking to improve branch and bound methods an application to open shop problems abstract only two branch and bound methods have been published so far for the open shop problem the best one has been proposed by brucker et al but some square problems from size 7 remain still unsolved with it we present an improving technique for branch and bound methods applied to brucker et al s algorithm for open shop problems our technique is based on intelligent backtracking an adaptation of a generic explanation system we have initially developed in the constraint programming scheme is used to develop that technique we tested our approach on open shop problems from the literature benchmarks of taillard the search is definitely improved on some square problems of size 10 the number of backtracks is reduced by more than 90 and we even solved an open problem of that size

a geometric approach to cut generating functions the cutting plane approach to integer programming was initiated more than 40 years ago gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and balas introduced intersection cuts for the corner polyhedron this line of research was left dormant for several decades until relatively recently when a paper of andersen louveaux weismantel and wolsey generated renewed interest in the corner polyhedron and intersection cuts recent developments rely on tools drawn from convex analysis geometry and number theory and constitute an elegant bridge between these areas and integer programming we survey these results and highlight recent breakthroughs in this area

an interior point constraint generation algorithm for semi infinite optimization with health care application we propose an interior point constraint generation ipcg algorithm for semi infinite linear optimization silo and prove that the algorithm converges to an e solution of silo after a finite number of constraints is generated we derive a complexity bound on the number of newton steps needed to approach the updated center after adding multiple violated constraints and a complexity bound on the total number of constraints that is required for the overall algorithm to converge we implement our algorithm to solve the sector duration optimization problem arising in leksell gamma knife perfexion elekta stockholm sweden treatment planning a highly specialized treatment for brain tumors using real patient data provided by the department of radiation oncology at princess margaret hospital in toronto ontario canada we show that our algorithm can efficiently handle problems in real life health care applications by providing a quality treatment plan in a timely manner comparing our computational results with mosek a commercial software package we show that the ipcg algorithm outperforms the classical primal dual interior point methods on sector duration optimization problem arising in perfexion treatment planning we also compare our results with that of a projected gradient method in both cases we show that ipcg algorithm obtains a more accurate solution substantially faster

minimal valid inequalities for integer constraints in this paper we consider a semi infinite relaxation of mixed integer linear programs we show that minimal valid inequalities for this relaxation correspond to maximal lattice free convex sets and that they arise from nonnegative piecewise linear positively homogeneous convex functions

conic mixed integer rounding cuts a conic integer program is an integer programming problem with conic constraints many problems in finance engineering statistical learning and probabilistic optimization are modeled using conic constraints here we study mixed integer sets defined by second order conic constraints we introduce general purpose cuts for conic mixed integer programming based on polyhedral conic substructures of second order conic sets these cuts can be readily incorporated in branch and bound algorithms that solve either second order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch and bound tree central to our approach is a reformulation of the second order conic constraints with polyhedral second order conic constraints in a higher dimensional space in this representation the cuts we develop are linear even though they are nonlinear in the original space of variables this feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed integer programming the reformulation also allows the use of polyhedral methods for conic integer programming we report computational results on solving unstructured second order conic mixed integer problems as well as mean variance capital budgeting problems and least squares estimation problems with binary inputs our computational experiments show that conic mixed integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed integer programs and hence improving their solvability

structure function scaling of bounded two dimensional turbulence statistical properties of forced two dimensional turbulence generated in two different flow domains are investigated by numerical simulations the considered geometries are the square domain and the periodic channel domain both bounded by lateral no slip sidewalls the focus is on the direct enstrophy cascade range and how the statistical properties change in the presence of no slip boundaries the scaling exponents of the velocity and the vorticity structure functions are compared to the classical kraichnan batchelor leith kbl theory which assumes isotropy homogeneity and self similarity for turbulence scales between the forcing and dissipation scale our investigation reveals that in the interior of the flow domain turbulence can be considered statistically isotropic and locally homogeneous for the enstrophy cascade range but it is weakly intermittent however the scaling of the vorticity structure function indicates a steeper slope for the energy spectrum than the kbl theory predicts near the walls the turbulence is strongly anisotropic at all flow scales

cut generating functions and s free sets we consider the separation problem for sets x that are pre images of a given set s by a linear mapping classical examples occur in integer programming as well as in other optimization problems such as complementarity one would like to generate valid inequalities that cut off some point not lying in x without reference to the linear mapping to this aim we introduce a concept cut generating functions cgf and we develop a formal theory for them largely based on convex analysis they are intimately related to s free sets and we study this relation disclosing several definitions for minimal cgf s and maximal s free sets our work unifies and puts into perspective a number of existing works on s free sets in particular we show how cgf s recover the celebrated gomory cuts

perspective cuts for a class of convex 0 1 mixed integer programs we show that the convex envelope of the objective function of mixed integer programming problems with a specific structure is the perspective function of the continuous part of the objective function using a characterization of the subdifferential of the perspective function we derive perspective cuts a family of valid inequalities for the problem perspective cuts can be shown to belong to the general family of disjunctive cuts but they do not require the solution of a potentially costly nonlinear programming problem to be separated using perspective cuts substantially improves the performance of branch cut approaches for at least two models that either naturally or after a proper reformulation have the required structure the unit commitment problem in electrical power production and the mean variance problem in portfolio optimization