Separable Integer Research Articles

Billiard partitions, Fibonacci sequences, SIP classes, and quivers

Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d d -dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B. We study the numbers of basis partitions with d d parts and relate them to the Fibonacci sequence and its natural generalizations. Remarkably, the generating series of basis partitions can be related to the quiver generating series of symmetric quivers corresponding to the framed unknot via knots-quivers correspondence, and to the count of Schröder paths.

Some separable integer partition classes

Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer partition classes with modulus [Formula: see text]. We also extend separable integer partition classes with modulus [Formula: see text] to overpartitions, called separable overpartition classes. We study overpartitions and the overpartition analogue of Rogers–Ramanujan identities, which are separable overpartition classes.

Modified Integer Sine Transforms. Construction Method and Separable Directional Adaptive Transforms for Intra Prediction in Image/Video Coding

The author proposes a matrix method for constructing a modified order 8 integer sine type VII transform. Based on the method, two modified order 8 integer sine type VII transforms are constructed and algorithms for fast implementation of 8-point modified integer sine type VII transforms are developed, which require only integer operations. These algorithms have low computational complexity that is 4.5 and 10.9 times less than that of the known algorithm. These transforms have higher coding performance in terms of quality and compression ratio than the known sine transforms do. Algorithms for fast implementation of 2D 8-point separable directional integer cosine and modified adaptive sine transforms for intra prediction with low computational complexity are developed, and their computational complexity is 4.62 and 8.24 times lower than that of the known algorithms.

Algorithms for Fast Implementation of 4-Point Integer Sine Type Vii Transformswithout Multiplication and Separable Directional Adaptive Transforms for Intra Prediction in Image/Video Coding

The separable directional adaptive integer cosine and sine type VII transforms with 4×4 blocks are proposed for intra prediction in image and video coding. Algorithms for fast implementation of 2D 4-point separable directional integer cosine and sine transforms are developed. These algorithms require 7 times fewer multiplication operations and provide higher compression ratio than the well-known algorithms. Two order-4 integer sine type VII transforms are proposed, and algorithms are developed for fast implementation of 4-point integer sine type VII transforms, which require integer operations only. As compared to the known algorithm, these algorithms can significantly reduce the number of multiplication operations or avoid them at all. These integer sine transforms are very close to discrete sine type VII transform and have a higher compression ratio than the well-known sine transform accepted in standard H.265 does.

Equilibrium Identification and Selection in IPGs

Problem definition: Integer programming games provide a framework to model (integer) decisions under competition. This has several applications ranging from decisions on competitive production volumes, over discrete capacity decisions among competitors to location selection under competition. The predominant solution concept for integer programming games in the literature is the identification of a Nash equilibrium; a strategy combination where no player benefits from unilaterally changing his/her strategy. While there are algorithms capable of identifying singular equilibria, they do not enumerate the full set of equilibria and thus cannot select the single most likely equilibrium. Methodology: Focusing on separable integer programming games, we propose a solution method in which we combine sampling techniques and equilibrium selection theory within an algorithm that integrates the determination of all equilibria and the subsequent identification of the most probable equilibrium among said equilibria. For this process, we use a column-generation approach, in which we divide the n-player integer programming game into a MIP-master problem capable of identifying equilibria in a sample, as well as column-generating subproblems tasked with the sampling of best responses and additional solution candidates. Results and managerial implications: We showcase algorithmic performance in various instances of knapsack games and large scale facility location and design games, including 3-player formulations. We highlight differences in solution quality between the proposed approach and the state of the art, enabling decision makers in competitive scenarios to base their actions on the most probable equilibrium rather than any identifiable equilibrium.

Distance confined path problem and separable integer programming

We consider in this article a problem of finding paths in a graph within a distance window, termed distance-confined path problem and develop a successive graph reduction scheme to solve such a problem effectively. Integrated with the surrogate constraint formulation, our proposed solution algorithm for the distance-confined path problem finds a prominent application in separable non-linear integer programming. By forming a family of distance-confined path problems, we are able to remove gradually all primal infeasible points in the region defined by the surrogate constraint and eventually achieve an exact equivalence between the surrogate constraint formulation and the primal problem, thus a zero duality gap.

Root Separation for Reducible Monic Quartics

We study root separation for reducible monic integer polynomials of degree four. If \text{H}(P) is the height and \mathrm{sep}(P) the minimal distance between two distinct roots of a separable integer polynomial P(x) , and \mathrm{sep}(P)=\mathrm{H}(P)^{-e(P)} , we show that \limsup e(P)=2 , where limsup is taken over all reducible monic integer polynomials P(x) of degree 4 .

Improving an exact approach for solving separable integer quadratic knapsack problems

We consider the specially structured (pure) integer Quadratic Multi-Knapsack Problem (QMKP) tackled in the paper "Exact solution methods to solve large scale integer quadratic knapsack problems" by D. Quadri, E. Soutif and P. Tolla (2009), recently appeared on this journal, where the problem is solved by transforming it into an equivalent 0---1 linearized Multi-Knapsack Problem (MKP). We show that, by taking advantage of the structure of the transformed (MKP), it is possible to derive an effective variable fixing procedure leading to an improved branch-and-bound approach. This procedure reduces dramatically the resulting linear problem size inducing an impressive improvement in the performances of the related branch and bound approach when compared to the results of the approach proposed by D. Quadri, E. Soutif and P. Tolla.

A roof linearization algorithm to obtain a tight upper bound for integer nonseparable quadratic programming

Abstract We study in this paper a general case of integer quadratic multi-knapsack problem (QMKP) where the objective function is non separable. An upper bound is proposed for (QMKP) which is computed via two steps. We first rewrite (QMKP) into an equivalent separable mixed integer quadratic program (QPx,y), using Gauss decomposition of the quadratic terms matrix. We then suggest an original technique, we call roof linearization, to linearize (QPx,y) so as to obtain a mixed integer linear program which optimal value provides an upper bound for both (QPx,y) and (QMKP). Preliminary computational experiments are conducted so as to assess that the proposed algorithm provides a tight upper bound in fast CPU times.

Separable Relaxation for Nonconvex Quadratic Integer Programming: Integer Diagonalization Approach

We present in this paper an integer diagonalization approach for deriving new lower bounds for general quadratic integer programming problems. More specifically, we introduce a semiunimodular transformation in order to diagonalize a symmetric matrix and preserve integral property of the feasible set at the same time. Via the semiunimodular transformation, the resulting separable quadratic integer program is a relaxation of the nonseparable quadratic integer program. We further define the integer diagonalization dual problem to identify the best semiunimodular transformation and analyze some basic properties of the set of semiunimodular transformations for a rational symmetric matrix. In particular, we present a complete characterization of the set of all semiunimodular transformations for a nonsingular 2×2 symmetric matrix. We finally discuss Lagrangian relaxation and convex relaxation schemes for the resulting separable quadratic integer programming problem and compare the tightness of different relaxation schemes.

Reducing the number of variables in integer quadratic programming problem

In this paper, a new variable reduction technique is presented for general integer quadratic programming problem ( GP ) , under which some variables of ( GP ) can be fixed at zero without sacrificing optimality. A sufficient condition and a necessary condition for the identification of dominated terms are provided. By comparing the given data of the problem and the upper bound of the variables, if they meet certain conditions, some variables can be fixed at zero. We report a computational study to demonstrate the efficacy of the proposed technique in solving general integer quadratic programming problems. Furthermore, we discuss separable integer quadratic programming problems in a simpler and clearer form.

Matrix decomposition and Lagrangian dual method for discrete portfolio optimization under concave transaction costs

In this paper, the discrete mean-variance model is considered for portfolio selection under concave transaction costs. By using the Cholesky decomposition technique, the convariance matrix to obtain a separable mixed integer nonlinear optimization problem is decomposed. A brand-and-bound algorithm based on Lagrangian relaxation is then proposed. Computational results are reported for test problems with the data randomly generated and those from the US stock market.

An exact algorithm for optimal redundancy in a series system with multiple component choices

In this paper, an exact algorithm was proposed for optimal redundancy in a series system with multiple component choices. A reformulation of the nonseparable reliability function was approximated by a separable integer programming problem. The resulting separable nonlinear integer programming problem is used to compute upper bounds by Lagrangian relaxation and dual search. A special partition scheme was derived to reduce the duality gap in a branch-and-bound process, thus ensure the convergence of the algorithm. Computational results show that the algorithm is efficient for solving this class of reliability optimization problems.

Convergent Lagrangian and Contour Cut Method for Nonlinear Integer Programming with a Quadratic Objective Function

In this paper we present an efficient exact solution method for solving nonlinear separable integer programming problems with a quadratic objective function. The proposed method combines the Lagrangian dual method with a duality reduction scheme using contour cut. At each iteration of the algorithm, lower and upper bounds of the problem are determined by the Lagrangian dual search. To eliminate the duality gap, a novel cut-and-partition scheme is derived by exploring the special structure of the quadratic contour. The method finds an exact solution of the problem in a finite number of iterations. Computational results are reported for problems with up to 2000 integer variables. Comparison results with other methods are also presented.

A method for determining optimal multipliers in the surrogate constraint method

It has recently been important to obtain optimal solutions to multidimensional nonlinear integer programming problems with multiple constraints. The surrogate constraint method is very effective in obtaining high-quality solutions of the original multidimensional problems. The surrogate constraint method translates a multidimensional problem into a one-dimensional problem by using a surrogate multiplier. But when there exists a surrogate duality gap between the translated one-dimensional problem and the original multidimensional problem, the optimal solution to the surrogate problem is not optimal with respect to the original problem. Nakagawa has recently proposed an improved surrogate constraint (ISC) method that can reduce the surrogate duality gap and provide an exact solution to the original problem. By using this method, we can obtain exact solutions to separable nonlinear integer programming problems with 1000 variables and 5 or 6 constraints. The ISC method requires an optimal surrogate multiplier. However, there are problems of calculation time and memory requirements in the algorithm for obtaining the optimal surrogate multiplier when the number of constraints is large. In this paper, the Dyer algorithm and Nakagawa's Cutting-Off Polyhedron (COP) algorithm for an algorithm that obtains the optimal surrogate multiplier in the ISC method are compared in computational experiments, and the advantages and disadvantages of these algorithms are investigated. In addition, an improved Dyer algorithm is proposed and computational experiments show that the improved algorithm is more effective than the original algorithm. © 2005 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 88(8): 38–48, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.20114

The values of Mahler measures

The set ℳ* of numbers which occur as Mahler measures of integer polynomials and the subset ℳ of Mahler measures of algebraic numbers (that is, of irreducible integer polynomials) are investigated. It is proved that every number α of degree d in ℳ* is the Mahler measure of a separable integer polynomial of degree at most with all its roots lying in the Galois closure F of ℚ(α), and every unit in ℳ is the Mahler measure of a unit in F of degree at most over ℚ This is used to show that some numbers considered earlier by Boyd are not Mahler measures. The set of numbers which occur as Mahler measures of both reciprocal and nonreciprocal algebraic numbers is also investigated. In particular, all cubic units in this set are described and it is shown that the smallest Pisot number is not the measure of a reciprocal number.

Mixed-model assembly line scheduling using the Lagrangian relaxation technique

The increasing market demand for product variety forces manufacturers to design mixed-model assembly lines on which different product models can be switched back and forth and mixed together with little changeover costs. This paper describes the design and implementation of an optimization-based scheduling algorithm for mixed-model compressor assembly lines at Toshiba with complicated component supply requirements. A separable integer optimization formulation is obtained by treating compressor lots going through a properly balanced line as undergoing a single operation, and the scheduling goal is to delivery products just in time while avoiding possible component shortage. The problem is solved b y using Lagrangian Relaxation (LR). Several generic defects of LR leading to slow algorithm convergence are identified based on geometrical insights, and are overcome by perturbing/ changing problem parameters. Numerical testing shows that near-optimal schedules are efficiently obtained, convergence is significantly improved, and the method is effective for practical problems. The system is currently under deployment at Toshiba

Scheduling flexible flow shops with sequence-dependent setup effects

This paper presents a Lagrangian relaxation-based approach for production scheduling of flexible flow shops (PSFFS) where sequence-dependent setup effects are significant. PSPFS is first formulated as a separable integer programming problem with synchronization constraints between production and machine usage. Lagrangian relaxation is then applied, and the scheduling is decomposed into part production and machine scheduling. In these subproblems, there are network flow structures in the equality constraints describing machine status change and production flow balance. Our iterative algorithm applies a minimum cost network flow algorithm to individual subproblems and adopts an efficient surrogate subgradient method to optimize Lagrangian multipliers. A machine availability-searching heuristic finally adjusts the solution to satisfy all synchronization constraints by exploiting the network structure, economic interpretation of Lagrangian multipliers, and the slack time policy. Numerical results of 16 cases, each having 20 test problems, demonstrate that differences between the schedules obtained by our approach and the true optima are on average within 15%, CPU times spent are all less than 17 min on a Pentium-II personal computer. Among the problem dimension factors, the number of part types has the most significant effect on both optimality and computation efficiency. Application of the methodology to daily scheduling of a realistic integrated circuit testing facility of 30 machines takes about 6 min of CPU time to generate a near-optimal solution.

Surrogate Gradient Algorithm for Lagrangian Relaxation

The subgradient method is used frequently to optimize dual functions in Lagrangian relaxation for separable integer programming problems. In the method, all subproblems must be solved optimally to obtain a subgradient direction. In this paper, the surrogate subgradient method is developed, where a proper direction can be obtained without solving optimally all the subproblems. In fact, only an approximate optimization of one subproblem is needed to get a proper surrogate subgradient direction, and the directions are smooth for problems of large size. The convergence of the algorithm is proved. Compared with methods that take effort to find better directions, this method can obtain good directions with much less effort and provides a new approach that is especially powerful for problems of very large size.

Refined proximity and sensitivity results in linearly constrained convex separable integer programming

We define a new measure of the structure of a linear constraint matrix and establish some propertie. We then use the measure to refine proximity and sensitivity results in the literature. In particular, we refine bounds on the distance between continuous and discrete optima of linearly constrained convex separable programming problems and on the size of the feasible region for a linear program. We apply the results to improve the computational complexity of Hochbaum and Shanthikumar's algorithm for integer nonlinear programming, reducing the computational effort for certain problems from exponential to polynomial.