Absolute value linear programming

We deal with linear programming problems involving absolute values in their formulations, so that they are no more expressible as standard linear programs. The presence of absolute values causes the problems to be nonconvex and nonsmooth, so hard to solve. In this paper, we study fundamental properties on the topology and the geometric shape of the solution set, and also conditions for convexity, connectedness, boundedness and integrality of the vertices. Further, we address various complexity issues, showing that many basic questions are NP-hard to solve. We show that the feasible set is a (nonconvex) polyhedral set and, more importantly, every nonconvex polyhedral set can be described by means of absolute value constraints. We also provide a necessary and sufficient condition when a KKT point of a nonconvex quadratic programming reformulation solves the original problem

Reformulations of mathematical programming problems as linear complementarity problems

A family of complementarity problems are defined as extensions of the well known Linear Complementarity Problem (LCP). These are (i.) Second Linear Complementarity Problem (SLCP) which is an LCP extended by introducing further equality restrictions and unrestricted variables, (ii.) Minimum Linear Complementarity Problem (MLCP) which is an LCP with additional variables not required to be complementary and with a linear objective function which is to be minimized, (iii.) Second Minimum Linear Complementarity Problem (SMLCP) which is an MLCP but the nonnegative restriction on one of each pair of complementary variables is relaxed so that it is allowed to be unrestricted in value. A number of well known mathematical programming problems, namely quadratic programming (convex, nonconvex, pseudoconvex nonconvex), bilinear programming, game theory, zero-one integer programming, the fixed charge problem, absolute value programming, variable separable programming are reformulated as members of this family of four complementarity problems

Statistical Mechanics Analysis of the Continuous Number Partitioning Problem

The number partitioning problem consists of partitioning a sequence of positive numbers ${a_1,a_2,..., a_N}$ into two disjoint sets, ${\cal A}$ and ${\cal B}$, such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. We use statistical mechanics tools to study analytically the Linear Programming relaxation of this NP-complete integer programming. In particular, we calculate the probability distribution of the difference between the cardinalities of ${\cal A}$ and ${\cal B}$ and show that this difference is not self-averaging.Comment: 9 pages, 1 figur

FPT-algorithms for some problems related to integer programming

In this paper, we present FPT-algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems' formulations are near square. The parameter is the maximum absolute value of rank minors of the corresponding matrices. Additionally, we present FPT-algorithms with respect to the same parameter for the problems, when the matrices have no singular rank sub-matrices.Comment: arXiv admin note: text overlap with arXiv:1710.00321 From author: some minor corrections has been don

Integer Programming and Incidence Treedepth

Recently a strong connection has been shown between the tractability of integer programming (IP) with bounded coefficients on the one side and the structure of its constraint matrix on the other side. To that end, integer linear programming is fixed-parameter tractable with respect to the primal (or dual) treedepth of the Gaifman graph of its constraint matrix and the largest coefficient (in absolute value). Motivated by this, Koutecký, Levin, and Onn [ICALP 2018] asked whether it is possible to extend these result to a more broader class of integer linear programs. More formally, is integer linear programming fixed-parameter tractable with respect to the incidence treedepth of its constraint matrix and the largest coefficient (in absolute value)? We answer this question in negative. We prove that deciding the feasibility of a system in the standard form, Ax=b,l≤x≤u , is NP-hard even when the absolute value of any coefficient in A is 1 and the incidence treedepth of A is 5. Consequently, it is not possible to decide feasibility in polynomial time even if both the assumed parameters are constant, unless P=NP

Numerical nonlinear inelastic analysis of stiffened shells of revolution. Volume 3: Engineer's program manual for STARS-2P digital computer program

Engineering programming information is presented for the STARS-2P (shell theory automated for rotational structures-2P (plasticity)) digital computer program, and FORTRAN 4 was used in writing the various subroutines. The execution of this program requires the use of thirteen temporary storage units. The program was initially written and debugged on the IBM 370-165 computer and converted to the UNIVAC 1108 computer, where it utilizes approximately 60,000 words of core. Only basic FORTRAN library routines are required by the program: sine, cosine, absolute value, and square root