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CPS - Linear and Integer Programming
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Laurence A. Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access. Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Get access to the full version of this article. View access options below. You previously purchased this article through ReadCube. Institutional Login. It consists of the following three parts:. The problem is usually expressed in matrix form , and then becomes:.
Other forms, such as minimization problems, problems with constraints on alternative forms, as well as problems involving negative variables can always be rewritten into an equivalent problem in standard form. Suppose that a farmer has a piece of farm land, say L km 2 , to be planted with either wheat or barley or some combination of the two. The farmer has a limited amount of fertilizer, F kilograms, and pesticide, P kilograms.
Every square kilometer of wheat requires F 1 kilograms of fertilizer and P 1 kilograms of pesticide, while every square kilometer of barley requires F 2 kilograms of fertilizer and P 2 kilograms of pesticide. Let S 1 be the selling price of wheat per square kilometer, and S 2 be the selling price of barley.
Linear and integer programming - theory and practice
If we denote the area of land planted with wheat and barley by x 1 and x 2 respectively, then profit can be maximized by choosing optimal values for x 1 and x 2. This problem can be expressed with the following linear programming problem in the standard form:. Linear programming problems can be converted into an augmented form in order to apply the common form of the simplex algorithm.
This form introduces non-negative slack variables to replace inequalities with equalities in the constraints. The problems can then be written in the following block matrix form:. Every linear programming problem, referred to as a primal problem, can be converted into a dual problem , which provides an upper bound to the optimal value of the primal problem.
In matrix form, we can express the primal problem as:. There are two ideas fundamental to duality theory.
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One is the fact that for the symmetric dual the dual of a dual linear program is the original primal linear program. Additionally, every feasible solution for a linear program gives a bound on the optimal value of the objective function of its dual. The weak duality theorem states that the objective function value of the dual at any feasible solution is always greater than or equal to the objective function value of the primal at any feasible solution.
A linear program can also be unbounded or infeasible. Duality theory tells us that if the primal is unbounded then the dual is infeasible by the weak duality theorem. Likewise, if the dual is unbounded, then the primal must be infeasible. However, it is possible for both the dual and the primal to be infeasible. See dual linear program for details and several more examples. A covering LP is a linear program of the form:. The dual of a covering LP is a packing LP , a linear program of the form:. Covering and packing LPs commonly arise as a linear programming relaxation of a combinatorial problem and are important in the study of approximation algorithms.
The LP relaxations of the set cover problem , the vertex cover problem , and the dominating set problem are also covering LPs. Finding a fractional coloring of a graph is another example of a covering LP. In this case, there is one constraint for each vertex of the graph and one variable for each independent set of the graph.
It is possible to obtain an optimal solution to the dual when only an optimal solution to the primal is known using the complementary slackness theorem. The theorem states:.
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Then x and y are optimal for their respective problems if and only if. So if the i -th slack variable of the primal is not zero, then the i -th variable of the dual is equal to zero. Likewise, if the j -th slack variable of the dual is not zero, then the j -th variable of the primal is equal to zero.
This necessary condition for optimality conveys a fairly simple economic principle. In standard form when maximizing , if there is slack in a constrained primal resource i. Likewise, if there is slack in the dual shadow price non-negativity constraint requirement, i. Geometrically, the linear constraints define the feasible region , which is a convex polyhedron. A linear function is a convex function , which implies that every local minimum is a global minimum ; similarly, a linear function is a concave function , which implies that every local maximum is a global maximum.
An optimal solution need not exist, for two reasons. Second, when the polytope is unbounded in the direction of the gradient of the objective function where the gradient of the objective function is the vector of the coefficients of the objective function , then no optimal value is attained because it is always possible to do better than any finite value of the objective function.
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Otherwise, if a feasible solution exists and if the constraint set is bounded, then the optimum value is always attained on the boundary of the constraint set, by the maximum principle for convex functions alternatively, by the minimum principle for concave functions since linear functions are both convex and concave. However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function that is, the constant function taking the value zero everywhere.
For this feasibility problem with the zero-function for its objective-function, if there are two distinct solutions, then every convex combination of the solutions is a solution. The vertices of the polytope are also called basic feasible solutions. The reason for this choice of name is as follows. Let d denote the number of variables. Thereby we can study these vertices by means of looking at certain subsets of the set of all constraints a discrete set , rather than the continuum of LP solutions.
This principle underlies the simplex algorithm for solving linear programs. The simplex algorithm , developed by George Dantzig in , solves LP problems by constructing a feasible solution at a vertex of the polytope and then walking along a path on the edges of the polytope to vertices with non-decreasing values of the objective function until an optimum is reached for sure.
In many practical problems, " stalling " occurs: many pivots are made with no increase in the objective function. In practice, the simplex algorithm is quite efficient and can be guaranteed to find the global optimum if certain precautions against cycling are taken. The simplex algorithm has been proved to solve "random" problems efficiently, i. However, the simplex algorithm has poor worst-case behavior: Klee and Minty constructed a family of linear programming problems for which the simplex method takes a number of steps exponential in the problem size. Like the simplex algorithm of Dantzig, the criss-cross algorithm is a basis-exchange algorithm that pivots between bases.
However, the criss-cross algorithm need not maintain feasibility, but can pivot rather from a feasible basis to an infeasible basis.
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The criss-cross algorithm does not have polynomial time-complexity for linear programming. In contrast to the simplex algorithm, which finds an optimal solution by traversing the edges between vertices on a polyhedral set, interior-point methods move through the interior of the feasible region. This is the first worst-case polynomial-time algorithm ever found for linear programming. To solve a problem which has n variables and can be encoded in L input bits, this algorithm uses O n 4 L pseudo-arithmetic operations on numbers with O L digits.
Leonid Khachiyan solved this long-standing complexity issue in with the introduction of the ellipsoid method. The convergence analysis has real-number predecessors, notably the iterative methods developed by Naum Z. Shor and the approximation algorithms by Arkadi Nemirovski and D. Khachiyan's algorithm was of landmark importance for establishing the polynomial-time solvability of linear programs.
The algorithm was not a computational break-through, as the simplex method is more efficient for all but specially constructed families of linear programs. However, Khachiyan's algorithm inspired new lines of research in linear programming. In , N.