35,657 research outputs found
The use of Grossone in Mathematical Programming and Operations Research
The concepts of infinity and infinitesimal in mathematics date back to
anciens Greek and have always attracted great attention. Very recently, a new
methodology has been proposed by Sergeyev for performing calculations with
infinite and infinitesimal quantities, by introducing an infinite unit of
measure expressed by the numeral grossone. An important characteristic of this
novel approach is its attention to numerical aspects. In this paper we will
present some possible applications and use of grossone in Operations Research
and Mathematical Programming. In particular, we will show how the use of
grossone can be beneficial in anti--cycling procedure for the well-known
simplex method for solving Linear Programming Problems and in defining exact
differentiable Penalty Functions in Nonlinear Programming
A double-pivot degenerate-robust simplex algorithm for linear programming
A double pivot algorithm that combines features of two recently published
papers by these authors is proposed. The proposed algorithm is implemented in
MATLAB. The MATLAB code is tested, along with a MATLAB implementation of
Dantzig's algorithm, for several test sets, including a set of cycling LP
problems, Klee-Minty's problems, randomly generated linear programming (LP)
problems, and Netlib benchmark problems. The test result shows that the
proposed algorithm is (a) degenerate-tolerance as we expected, and (b) more
efficient than Dantzig's algorithm for large size randomly generated LP
problems but less efficient for Netlib benchmark problems and small size
randomly generated problems in terms of CPU time.Comment: 21 pages, 1 figure, and 2 table
The s-monotone index selection rules for pivot algorithms of linear programming
In this paper we introduce the concept of s-monotone index selection rule for linear programming problems. We show that several known anti-cycling pivot rules like the minimal index, Last-InâFirst-Out and the most-often-selected-variable pivot rules are s-monotone index selection rules. Furthermore, we show a possible way to define new s-monotone pivot rules. We prove that several known algorithms like the primal (dual) simplex, MBU-simplex algorithms and criss-cross algorithm with s-monotone pivot rules are finite methods. We implemented primal simplex and primal MBU-simplex algorithms, in MATLAB, using three s-monotone index selection rules, the minimal-index, the Last-InâFirst-Out (LIFO) and the Most-Often-Selected-Variable (MOSV) index selection rule. Numerical results demonstrate the viability of the above listed s-monotone index selection rules in the framework of pivot algorithms
Suboptimal Solution Path Algorithm for Support Vector Machine
We consider a suboptimal solution path algorithm for the Support Vector
Machine. The solution path algorithm is an effective tool for solving a
sequence of a parametrized optimization problems in machine learning. The path
of the solutions provided by this algorithm are very accurate and they satisfy
the optimality conditions more strictly than other SVM optimization algorithms.
In many machine learning application, however, this strict optimality is often
unnecessary, and it adversely affects the computational efficiency. Our
algorithm can generate the path of suboptimal solutions within an arbitrary
user-specified tolerance level. It allows us to control the trade-off between
the accuracy of the solution and the computational cost. Moreover, We also show
that our suboptimal solutions can be interpreted as the solution of a
\emph{perturbed optimization problem} from the original one. We provide some
theoretical analyses of our algorithm based on this novel interpretation. The
experimental results also demonstrate the effectiveness of our algorithm.Comment: A shorter version of this paper is submitted to ICML 201
An Active Set Algorithm for Robust Combinatorial Optimization Based on Separation Oracles
We address combinatorial optimization problems with uncertain coefficients
varying over ellipsoidal uncertainty sets. The robust counterpart of such a
problem can be rewritten as a second-oder cone program (SOCP) with integrality
constraints. We propose a branch-and-bound algorithm where dual bounds are
computed by means of an active set algorithm. The latter is applied to the
Lagrangian dual of the continuous relaxation, where the feasible set of the
combinatorial problem is supposed to be given by a separation oracle. The
method benefits from the closed form solution of the active set subproblems and
from a smart update of pseudo-inverse matrices. We present numerical
experiments on randomly generated instances and on instances from different
combinatorial problems, including the shortest path and the traveling salesman
problem, showing that our new algorithm consistently outperforms the
state-of-the art mixed-integer SOCP solver of Gurobi
MICROCOMPUTER ACCURACY IN SOLVING LINEAR PROGRAMMING PROBLEMS WITH REDUNDANT CONSTRAINTS
This study reports on how different microcomputer systems performed in the solution of two linear programming models purposely specified with redundant vectors. Comparisons were made to a Cyber 720 that used both a Fortran and Basic version of the same primal-dual algorithm. Results are mixed. But Microsoft Basic with double precision under CP/M on a Z80A processor performed at least equally well to the Cyber 720 provided that an appropriate essential zero value was specified. Different coefficient scaling schemes were also tested. The results should be of interest to all users of matrix inversion schemes on microcomputers. Extensions of the study to new hardware and software systems are encouraged.Research Methods/ Statistical Methods,
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