Question

Interger Programming and Nonlinear Programing are two important extensions of the basic Simplex Linear Programing model. How does each differ from Simplex in theory and practice? How does Solver adjust for these two extensions? What about Rosenbrock’s method, cutting plane procedure, and interior point methods?

Answer 1

An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear.

Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems.

If some decision variables are not discrete the problem is known as a mixed-integer programming problem.

Canonical and standard form for ILPs

An integer linear program in canonical form is expressed as:

and an ILP in standard form is expressed as

where are vectors and is a matrix, where all entries are integers. As with linear programs, ILPs not in standard form can be conv erted to standard form by eliminating inequalities, introducing slack variables and replacing variables that are not sign-constrained with the difference of two sign-constrained variables

Nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or stationary points) of an objective function over a set of unknown real variables and conditional to the satisfaction of a system of equalities and inequalities, collectively termed constraints. It is the sub-field of mathematical optimization that deals with problems that are not linear.

Let n, m, and p be positive integers. Let X be a subset of Rⁿ, let f, g_i, and h_j be real-valued functions on X for each i in {1, …, m} and each j in {1, …, p}, with at least one of f, g_i, and h_j being nonlinear.

A nonlinear minimization problem is an optimization problem of the form

A nonlinear maximization problem is defined in a similar way.

Rosenbrock search is a numerical optimization algorithm applicable to optimization problems in which the objective function is inexpensive to compute and the derivative either does not exist or cannot be computed efficiently.

The basic idea of the cutting plane method is to cut off parts of the feasible region of the LP relaxation, so that the optimal integer solution becomes an extreme point and therefore can be found by the simplex method.

Interior-point methods (also referred to as barrier methods or IPMs) are a certain class of algorithms that solve linear and nonlinear convex optimization problems.

Note: Plzzz don' t give dislike.....Plzzz comment if u have any problem i will try to resolve it.......