Theory of Linear and Integer Programming

The symmetric groups Sn and the cyclic groups Cn essentially are the only examples for symmetry groups of linear or integer programs that have been discussed in the literature, see e.g. [5] and [6]. In [4], Bodi, Herr, and Joswig developed some ideas to tackle linear and integer programs with arbitrary groups of symmetries. However, the question remained whether or not there are linear (integer) programs with groups of symmetries other than Sn and Cn. Indeed, we show in this short note that every finite permutation group is the full symmetry group of a suitable linear or integer program. Some of our constructions are based on graph theory.

Previous article Next article Complementarity Theorems for Linear ProgrammingA. C. WilliamsA. C. Williamshttps://doi.org/10.1137/1012015PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Frank Eugene Clark, Mathematical Notes: Remark on the Constraint Sets in Linear Programming, Amer. Math. Monthly, 68 (1961), 351–352 MR1531192 0109.38204 CrossrefISIGoogle Scholar[2] A. J. Goldman, Resolution and separation theorems for polyhedral convex setsLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N. J., 1956, 41–51, in [4] MR0089113 0072.37505 Google Scholar[3] A. J. Goldman and , A. W. Tucker, Theory of linear programmingLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., 1956, 53–97, in [4] MR0101826 0072.37601 Google Scholar[4] H. W. Kuhn and , A. W. Tucker, Linear Inequalities and Related Systems, Princeton University Press, Princeton, 1956 0072.37502 Google Scholar[5] A. C. Williams, Boundedness relations for linear constraint sets, Linear Algebra and Appl., 3 (1970), 129–141 10.1016/0024-3795(70)90009-1 MR0266612 0201.22003 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Sparse solutions to an underdetermined system of linear equations via penalized Huber loss6 November 2020 | Optimization and Engineering, Vol. 22, No. 3 Cross Ref Dynamic Non-diagonal Regularization in Interior Point Methods for Linear and Convex Quadratic Programming26 February 2019 | Journal of Optimization Theory and Applications, Vol. 181, No. 3 Cross Ref Necessary and Sufficient Conditions for Noiseless Sparse Recovery via Convex Quadratic SplinesMustafa Ç Pinar12 February 2019 | SIAM Journal on Matrix Analysis and Applications, Vol. 40, No. 1AbstractPDF (416 KB)A labeling algorithm for the sensitivity ranges of the assignment problemApplied Mathematical Modelling, Vol. 35, No. 10 Cross Ref Bibliography15 August 2011 Cross Ref Sensitivity Analysis and Dynamic Programming15 February 2011 Cross Ref Bibliography Cross Ref Condition measures and properties of the central trajectory of a linear programMathematical Programming, Vol. 83, No. 1-3 Cross Ref New characterizations of ℓ1 solutions to overdetermined systems of linear equationsOperations Research Letters, Vol. 16, No. 3 Cross Ref Limiting behavior of weighted central paths in linear programmingMathematical Programming, Vol. 65, No. 1-3 Cross Ref Stability of linearly constrained convex quadratic programsJournal of Optimization Theory and Applications, Vol. 64, No. 1 Cross Ref Marginal values in mixed integer linear programmingMathematical Programming, Vol. 44, No. 1-3 Cross Ref A theory of linear inequality systemsLinear Algebra and its Applications, Vol. 106 Cross Ref Boundedness relations in linear semi-infinite programmingAdvances in Applied Mathematics, Vol. 8, No. 1 Cross Ref A Variable-Complexity Norm Maximization ProblemO. L. Mangasarian and T. -H. Shiau17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 7, No. 3AbstractPDF (698 KB)Simple computable bounds for solutions of linear complementarity problems and linear programs26 February 2009 Cross Ref On polyhedral extension of some LP theoremsMathematical Programming, Vol. 30, No. 1 Cross Ref Polyhedral extensions of some theorems of linear programmingMathematical Programming, Vol. 24, No. 1 Cross Ref Optimal simplex tableau characterization of unique and bounded solutions of linear programsJournal of Optimization Theory and Applications, Vol. 35, No. 1 Cross Ref Projection and Restriction Methods in Geometric Programming and Related Problems Cross Ref Representation of Convex Sets Cross Ref Projection and restriction methods in geometric programming and related problemsJournal of Optimization Theory and Applications, Vol. 26, No. 1 Cross Ref The complementary unboundedness of dual feasible solution sets in convex programmingMathematical Programming, Vol. 12, No. 1 Cross Ref Theorems on the dimension of convex setsLinear Algebra and its Applications, Vol. 12, No. 1 Cross Ref On the primal and dual constraint sets in geometric programmingJournal of Mathematical Analysis and Applications, Vol. 32, No. 3 Cross Ref Volume 12, Issue 1| 1970SIAM Review History Submitted:23 December 1968Published online:18 July 2006 InformationCopyright © 1970 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1012015Article page range:pp. 135-137ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Job shops are an important production environment for low-volume high-variety manufacturing. Its scheduling has recently been formulated as an integer linear programming (ILP) problem to take advantages of popular mixed-integer linear programming (MILP) methods, e.g., branch-and-cut. When considering a large number of parts, MILP methods may experience difficulties. To address this, a critical but much overlooked issue is formulation tightening. The idea is that if problem constraints can be transformed to directly delineate the problem convex hull in the data preprocessing stage, then a solution can be obtained by using linear programming (LP) methods without combinatorial difficulties. The tightening process, however, is fundamentally challenging because of the existence of integer variables. In this article, an innovative and systematic approach is established for the first time to tighten the formulations of individual parts, each with multiple operations, in the data preprocessing stage. It is a major advancement of our previous work on problems with binary and continuous variables to integer variables. The idea is to first link integer variables to binary variables by innovatively combining constraints so that the integer variables are uniquely determined by the binary variables. With binary and continuous variables only, it is proved that the vertices of the convex hull can be obtained based on vertices of the LP problem after relaxing binary requirements. These vertices are then converted to tightened constraints for general use. This approach significantly improves our previous results on tightening individual operations. Numerical results demonstrate significant benefits on solution quality and computational efficiency. This approach also applies to other complex ILP and MILP problems with similar characteristics and fundamentally changes the way how such problems are formulated and solved. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Scheduling is an important but difficult problem in planning and operation of job shops. The problem has been recently formulated in an integer linear programming (ILP) form to take advantage of popular mixed-integer linear programming methods. Given an ILP problem, there must exist a linear programming (LP) formulation so that all of its vertices are also the vertices to the ILP problem. If such an LP problem can be found in the data preprocessing stage, then the corresponding ILP problem is tight and can be solved by using an LP method without difficulties. In this article, an innovative and systematic approach is established to tighten the formulations of individual parts, each with one or multiple operations. It is a major advancement of our previous work on problems with binary and continuous variables by novel exploitation of the relationship between integer and binary variables in job-shop scheduling. The resulting tightened constraints are characterized by part parameters and the length of the scheduling horizon and can be easily adjusted for other data sets. Results demonstrate significant benefits on solution quality and computational efficiency. This approach also applies to other complex ILP and MILP problems with similar characteristics and fundamentally changes the way how such problems are formulated and solved.

Whole Life Cost Comparisons Based Upon the Year of Required Protection ABSTRACT The traditional measures of Interest Adjusted Surrender Cost and Linton's rate of return are computed for 68 whole life policies. Similar measures are obtained via linear programming where cash flows are discounted using several different external interest rates. With the linear programming method, year of insurance protection is varied to include year 0, year 10 and year 20. If protection is required in year 20, several policies are infeasible (incapable of generating any insurance protection). Upper limits for rates of return are calculated. Infeasibility occurs for a given policy if the external rate of return exceeds the upper limit. Introduction The purpose of this article is to use a linear programming (LP) method for measuring the cost of whole life insurance. The method is applied to policies offered in 1984 by 68 different insurers. For comparative purposes, the traditional methods of interest adjusted surrender cost (IASC) and Linton's rate of return are applied to the same set of policies. The insurers and policies were selected using criteria established in an earlier study by Hutchins and Quenneville [3]. Consequently, the results of this study of 1984 policies may be compared to a set of similar policies offered in 1972. The LP method is similar to both IASC and Linton methods in several respects. Specifically, all three methods assume deterministic projected dividends and a fixed horizon of 20 years. The LP technique can be used to derive level interest adjusted costs similar to the IASC and internal rates of return on equity similar to the Linton rate of return. Nevertheless, both the IASC method and Linton's method rely on assumptions not required by the LP method. The LP method requires only an assumption of a rate of return that is relevant to the policyholder. Conceptually, the IASC and Linton's method treat the whole life insurance policy as providing the two products of protection and savings. The IASC attempts to measure the level cost of protection while the Linton method attempts to measure the rate of return on the savings component. The IASC requires an interest rate for discounting whereas Linton's method requires various term insurance rates. As a consequence of these different required assumptions, rankings obtained by these two methods will not be perfectly correlated. The LP method, by contrast, does not attempt to directly separate protection and savings. The method assumes that the insured individual requires a given level of protection. It is irrelevant to the insured how the insurer provides the protection. In other words, the insured is unconcerned with how the insurer divides the premium into loading charges, reserves, and so forth. The LP method has the additional flexibility of considering the point in time at which the insured requires protection. Neither the IASC nor Linton method explicitly address issues related to timing of insurance protection. The flexibility of varying the year of required protection is the primary characteristic of the LP method that differentiates it from the traditional methods. The IASC, for example, implicitly assumes that coverage is required at the time the policy becomes effective. The IASC, however, does not recognize the reduction in the insured's wealth resulting from the premium payment. The LP method recognizes premium payments by increasing the policy face value by a corresponding amount. With the LP method, premiums are compounded and accumulated at the relevant interest rate to the year in which protection is required. This accumulation of premiums may be large if protection is required 15 or 20 years into the future. For large interest rates the accumulated premiums will, at some point, exceed the face value of the policy. …

Apart from Jungnickel’s book [17] mentioned in the preface, we suggest (in alphabetical order of the first author’s surname) the following textbooks: 1. Linear Programming by V. Chvatal [11] This is one of the best introductions on linear programming and optimization. Most of the concepts and techniques are introduced using illustrative examples. 2. Introduction to Operations Research by F.J. Hillier and G.J. Lieberman [15] This is a standard work on general operations research. 3. Operations Research; Models and Methods by P.A. Jensen and J.F. Bard [16] This is again a standard reference on general operations research. 4. Linear Optimization and Extensions by J.M. Padberg [22] This book on linear optimization grew out of a series of lectures and contains a chapter that relates combinatorial optimization with linear optimization. 5. Theory of Linear and Integer Programming by A. Schrijver [24] This book provides a thorough mathematical treatment of the theory of linear and integer optimization. It is aimed at an advanced level. 6. Linear and Integer Programming; Theory and Practice by G. Sierksma [26] This book on linear programming/optimization contains a large number of case studies, including several on network optimization. 7. Operations Research: an Introduction by H.A. Taha [29] This is another standard work on general operations research. 8. Model Building in Mathematical Programming, and Model Solving in Mathematical Programming by H.P. Williams [30, 31] These two books on building and solving mathematical models are very well written, and cover aspects of model building and solving that are dealt with very cursorily in other introductory books. The first book also contains a number of case studies on mathematical modeling. 9. Operations Research: Applications and Algorithms by W.L. Winston [33] This again is a major standard work on general operations research.

Engineering Optimization Theory and Practice

Motivated by Bland's linear programming (LP) generalization of the renowned Edmonds--Karp efficient refinement of the Ford--Fulkerson maximum flow algorithm, we analyze three closely related natural augmentation rules for LP and integer-linear programming (ILP) augmentation algorithms. For all three rules and in both contexts, LP and ILP, we bound the number of augmentations. Extending Bland's “discrete steepest-descent” augmentation rule (i.e., choosing directions with the best ratio of cost improvement per unit 1-norm length, and then making maximal augmentations in such directions) from LP to ILP, we (i) show that the number of discrete steepest-descent augmentations is bounded by the number of elements in the Graver basis of the problem matrix and (ii) give the first strongly polynomial-time algorithm for $N$-fold ILP. For LP, two of the rules can suffer from a “zig-zagging” phenomenon, and so in those cases we apply the rules more subtly to achieve good bounds. Our results improve on what is known for such augmentation algorithms for LP (e.g., extending the style of bounds found by Kitahara and Mizuno for the number of steps in the simplex method) and are closely related to research on the diameters of polytopes and the search for a strongly polynomial-time simplex or augmentation algorithm.

A common type of mathematical optimization is Linear Programming (LP). An LP solution of aquifer influence functions has recently been reported by Gadjica, etal.1 (1987) and Targac, etal.2 Their LP matrices were large and sparse (only 3% of the elements were non-zero) and were solved on main frame computers. Another recent application of LP is equation-of-state matching of laboratory PVT data3. This problem leads to a smaller, denser LP matrix. Three methods of LP solution were investigated on microcomputers: (1) the simplex method, (2) the revised simplex method, and (3) the symmetric method. These methods were run on several LP problems ranging from a small dense matrix to large sparse matrices. The different methods have different characteristics which affect the speed, storage requirements and simplicity of coding. The simplex method is straightforward, but usually is slower and requires more storage than the other methods. The results of this study are tabulated with running times and storage requirements for the various LP methods and microcomputers. The computers range from the IBM XT to the Compaq 386. This information serves as a documentation of the LP codes and should be useful for an engineer interested in using LP codes on a microcomputer.

In an earlier work, Sun and Bayen built a Large-Capacity Cell Transmission Model for air traffic flow management. They formulated an integer programming problem of minimizing the total travel time of flights in the National Airspace System of the United States subject to sector capacity constraints. The integer programming was relaxed to a linear programming for computational efficiency. In this paper the authors formulate the optimization problem in a standard linear programming form. We analyze the total unimodular property of the constraint matrix, and prove that the linear programming relaxation generates an integral optimal solution for the original integer programming. It is guaranteed to be optimal and integral if solved by the simplex method. Furthermore, we find the degeneracies for both feasible polyhedron and optimal polyhedron. In order to speed up the computation, we present a revised Dantzig-Wolfe Decomposition algorithm, which is shown to preserve the total unimodularity of the constraint matrix and successfully resolve the degeneracies.

Hybrid-LP: Finding advanced starting points for simplex, and pivoting LP methods

complementarity, fixed points, solving equations, piecewise linear, path following, convex polyhedra

A Survey of Lagrangean Techniques for Discrete Optimization

For many of us, modern-day linear programming (LP) started with the work of George Dantzig in 1947. However, it must be said that many other scientists have also made seminal contributions to the subject, and some would argue that the origins of LP predate Dantzig’s contribution. It is matter open to debate [36]. However, what is not open to debate is Dantzig’s key contribution to LP computation. In contrast to the economists of his time, Dantzig viewed LP not just as a qualitative tool in the analysis of economic phenomena, but as a method that could be used to compute actual answers to specific real-world problems. Consistent with that view, he proposed an algorithm for solving LPs, the simplex algorithm [12]. To this day the simplex algorithm remains a primary computational tool in linear and mixed-integer programming (MIP). In [11] it is reported that the first application of Dantzig’s simplex algorithm to the solution of a non-trivial LP was Laderman’s solution of a 21 constraint, 77 variable instance of the classical Stigler Diet Problem [41]. It is reported that the total computation time was 120 man-days! The first computer implementation of an at-least modestly general version of the simplex algorithm is reported to have been on the SEAC computer at the then National Bureau of Standards [25]. (There were apparently some slightly earlier implementations for dealing with models that were “triangular”, that is, where all the linear systems could be solved by simple addition and subtraction.) Orchard-Hays [35] reports that several small instances having as many as 10 constraints and 20 variables were solved with this implementation. The first systematic development of computer codes for the simplex algorithm began very shortly thereafter at the RAND Corporation in Santa Monica, California. Dantzig’s initial LP work occurred at the Air Force following

Introduction to mathematical programming: N. K. Kwak and Marc J. Schniederjans: Krieger, Malabar

Operations Research: Applications and Algorithms