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  4. Some Linear Programming References

Perhaps my expectations were unrealistic. I have some experience with linear programming, and was hoping to leverage that knowledge to learn about economic analysis. Often "classics" are at least well written; it wasn't that either. The writing was scattered and repetitive: This despite the lovely quote " Their peculiar use of "efficiency" confused me until they defined it a hundred pages later.

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It consists of the following three parts:. The problem is usually expressed in matrix form , and then becomes:.

Linear programming

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.

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. 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. As an example, consider the linear program:. Revisit the above example of the farmer who may grow wheat and barley with the set provision of some L land, F fertilizer and P pesticide.

Assume now that y unit prices for each of these means of production inputs are set by a planning board. The planning board's job is to minimize the total cost of procuring the set amounts of inputs while providing the farmer with a floor on the unit price of each of his crops outputs , S 1 for wheat and S 2 for barley.

This corresponds to the following linear programming problem:. The primal problem deals with physical quantities. With all inputs available in limited quantities, and assuming the unit prices of all outputs is known, what quantities of outputs to produce so as to maximize total revenue? The dual problem deals with economic values. With floor guarantees on all output unit prices, and assuming the available quantity of all inputs is known, what input unit pricing scheme to set so as to minimize total expenditure? To each variable in the primal space corresponds an inequality to satisfy in the dual space, both indexed by output type.

To each inequality to satisfy in the primal space corresponds a variable in the dual space, both indexed by input type.

Linear Programming

The coefficients that bound the inequalities in the primal space are used to compute the objective in the dual space, input quantities in this example. The coefficients used to compute the objective in the primal space bound the inequalities in the dual space, output unit prices in this example.

Both the primal and the dual problems make use of the same matrix. In the primal space, this matrix expresses the consumption of physical quantities of inputs necessary to produce set quantities of outputs. In the dual space, it expresses the creation of the economic values associated with the outputs from set input unit prices. Since each inequality can be replaced by an equality and a slack variable, this means each primal variable corresponds to a dual slack variable, and each dual variable corresponds to a primal slack variable. This relation allows us to speak about complementary slackness.

Sometimes, one may find it more intuitive to obtain the dual program without looking at the program matrix. Consider the following linear program:.


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Since this is a minimization problem, we would like to obtain a dual program that is a lower bound of the primal. In other words, we would like the sum of all right hand side of the constraints to be the maximal under the condition that for each primal variable the sum of its coefficients do not exceed its coefficient in the linear function. This sum must be at most c 1.


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As a result, we get:. Note that we assume in our calculations steps that the program is in standard form. However, any linear program may be transformed to standard form and it is therefore not a limiting factor.

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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.

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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.

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. First, if two constraints are inconsistent, then no feasible solution exists: 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.

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: 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. 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. Karmarkar proposed a projective method for linear programming. Karmarkar's algorithm improved on Khachiyan's worst-case polynomial bound giving O n 3. Karmarkar claimed that his algorithm was much faster in practical LP than the simplex method, a claim that created great interest in interior-point methods.

Affine scaling is one of the oldest interior point methods to be developed. It was developed in the Soviet Union in the mids, but didn't receive much attention until the discovery of Karmarkar's algorithm, after which affine scaling was reinvented multiple times and presented as a simplified version of Karmarkar's. Affine scaling amounts to doing gradient descent steps within the feasible region, while rescaling the problem to make sure the steps move toward the optimum faster.

For both theoretical and practical purposes, barrier function or path-following methods have been the most popular interior point methods since the s. The current opinion is that the efficiencies of good implementations of simplex-based methods and interior point methods are similar for routine applications of linear programming. Covering and packing LPs can be solved approximately in nearly-linear time.

There are several open problems in the theory of linear programming, the solution of which would represent fundamental breakthroughs in mathematics and potentially major advances in our ability to solve large-scale linear programs. This closely related set of problems has been cited by Stephen Smale as among the 18 greatest unsolved problems of the 21st century. In Smale's words, the third version of the problem "is the main unsolved problem of linear programming theory.

Some Linear Programming References

The development of such algorithms would be of great theoretical interest, and perhaps allow practical gains in solving large LPs as well. Although the Hirsch conjecture was recently disproved for higher dimensions, it still leaves the following questions open. These questions relate to the performance analysis and development of simplex-like methods. The immense efficiency of the simplex algorithm in practice despite its exponential-time theoretical performance hints that there may be variations of simplex that run in polynomial or even strongly polynomial time.

It would be of great practical and theoretical significance to know whether any such variants exist, particularly as an approach to deciding if LP can be solved in strongly polynomial time. The simplex algorithm and its variants fall in the family of edge-following algorithms, so named because they solve linear programming problems by moving from vertex to vertex along edges of a polytope. This means that their theoretical performance is limited by the maximum number of edges between any two vertices on the LP polytope.

As a result, we are interested in knowing the maximum graph-theoretical diameter of polytopal graphs. It has been proved that all polytopes have subexponential diameter. The recent disproof of the Hirsch conjecture is the first step to prove whether any polytope has superpolynomial diameter.

If any such polytopes exist, then no edge-following variant can run in polynomial time. Questions about polytope diameter are of independent mathematical interest. Simplex pivot methods preserve primal or dual feasibility. On the other hand, criss-cross pivot methods do not preserve primal or dual feasibility—they may visit primal feasible, dual feasible or primal-and-dual infeasible bases in any order.