linear programming

linear programming

linear programming, solution of a mathematical problem concerning maximum and minimum values of a first-degree (linear) algebraic expression, with variables subject to certain stated conditions (restraints). For example, the problem might be to find the minimum value of the expression x+y subject to the restraints x≥0, y≥0, 2x+y≥12, 5x+8y≥74, and x+6y≥24. The solution was set forth by the Russian mathematician L. V. Kantorovich in 1939 and was developed independently by the American George B. Dantzig, whose first work on the subject appeared in 1947. A faster, but more complex technique, that is suitable for problems with hundreds or thousands of variables, was developed by Bell Laboratories mathematician Naranda Karmarkar in 1983. Linear programming is particularly important in military and industrial planning.

Mathematical modeling technique useful for guiding quantitative decisions in business, industrial engineering, and to a lesser extent the social and physical sciences. Solving a linear programming problem can be reduced to finding the optimum value (see optimization) of a linear equation (called an objective function), subject to a set of constraints expressed as inequalities. The number of inequalities and variables depends on the complexity of the problem, whose solution is found by solving the system of inequalities like a system of equations. The extensive use of linear programming during World War II to deal with transportation, scheduling and allocations of resources under constraints like cost and priority gave the subject an impetus that carried it into the postwar era. The number of equations and variables needed to model real-life situations accurately is large, and the solution process can be time-consuming even with computers. Seealso simplex method.

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In mathematics, linear programming (LP) is a technique for optimization of a linear objective function, subject to linear equality and linear inequality constraints. Informally, linear programming determines the way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model given some list of requirements represented as linear equations.

More formally, given a polytope (for example, a polygon or a polyhedron), and a real-valued affine function

f(x_1, x_2, dots, x_n)=c_1x_1+c_2x_2+cdots +c_nx_n+d,

defined on this polytope, a linear programming method will find a point in the polytope where this function has the smallest (or largest) value. Such points may not exist, but if they do, searching through the polytope vertices is guaranteed to find at least one of them.

Linear programs are problems that can be expressed in canonical form:

Maximize mathbf{c}^T mathbf{x}
Subject to Amathbf{x} leq mathbf{b}

mathbf{x} represents the vector of variables (to be determined), while mathbf{c} and mathbf{b} are vectors of (known) coefficients and mathbf{A} is a (known) matrix of coefficients. The expression to be maximized or minimized is called the objective function (mathbf{c}^T mathbf{x} in this case). The equations Amathbf{x} leq mathbf{b} are the constraints which specify a convex polyhedron over which the objective function is to be optimized.

Linear programming can be applied to various fields of study. Most extensively it is used in business and economic situations, but can also be utilized for some engineering problems. Some industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design.

History of linear programming

The problem of solving a system of linear inequalities dates back at least as far as Fourier, after whom the method of Fourier-Motzkin elimination is named. Linear programming arose as a mathematical model developed during the second world war to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. It was kept secret until 1947. Postwar, many industries found its use in their daily planning.

The founders of the subject are George B. Dantzig, who published the simplex method in 1947, John von Neumann, who developed the theory of the duality in the same year, and Leonid Kantorovich, a Russian mathematician who used similar techniques in economics before Dantzig and won the Nobel prize in 1975 in economics. The linear programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger major theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior point method for solving linear programming problems.

Dantzig's original example of finding the best assignment of 70 people to 70 jobs exemplifies the usefulness of linear programming. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the number of particles in the universe. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the simplex algorithm. The theory behind linear programming drastically reduces the number of possible optimal solutions that must be checked.

Uses

Linear programming is an important field of optimization for several reasons. Many practical problems in operations research can be expressed as linear programming problems. Certain special cases of linear programming, such as network flow problems and multicommodity flow problems are considered important enough to have generated much research on specialized algorithms for their solution. A number of algorithms for other types of optimization problems work by solving LP problems as sub-problems. Historically, ideas from linear programming have inspired many of the central concepts of optimization theory, such as duality, decomposition, and the importance of convexity and its generalizations. Likewise, linear programming is heavily used in microeconomics and business management, either to maximize the income or minimize the costs of a production scheme. Some examples are food blending, inventory management, portfolio and finance management, resource allocation for human and machine resources, planning advertisement campaigns, etc.

Standard form

Standard form is the usual and most intuitive form of describing a linear programming problem. It consists of the following three parts:

  • A linear function to be maximized

e.g. maximize c_1 x_1 + c_2 x_2,

  • Problem constraints of the following form

e.g. a_{11} x_1 + a_{12} x_2 le b_1
a_{21} x_1 + a_{22} x_2 le b_2
a_{31} x_1 + a_{32} x_2 le b_3

  • Non-negative variables

e.g. x_1 ge 0
x_2 ge 0

The problem is usually expressed in matrix form, and then becomes:

maximize mathbf{c}^T mathbf{x}
subject to mathbf{A}mathbf{x} le mathbf{b}, , mathbf{x} ge 0

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.

Example

Suppose that a farmer has a piece of farm land, say A square kilometres large, to be planted with either wheat or barley or some combination of the two. The farmer has a limited permissible amount F of fertilizer and P of insecticide which can be used, each of which is required in different amounts per unit area for wheat (F1, P1) and barley (F2, P2). Let S1 be the selling price of wheat, and S2 the price of barley. If we denote the area planted with wheat and barley by x1 and x2 respectively, then the optimal number of square kilometres to plant with wheat vs barley can be expressed as a linear programming problem:
maximize S_1 x_1 + S_2 x_2 , (maximize the revenue — revenue is the "objective function")
subject to x_1 + x_2 le A (limit on total area)
F_1 x_1 + F_2 x_2 le F (limit on fertilizer)
P_1 x_1 + P_2 x_2 le P (limit on insecticide)
x_1 ge 0,, x_2 ge 0 (cannot plant a negative area)

Which in matrix form becomes:

maximize begin{bmatrix} S_1 & S_2 end{bmatrix} begin{bmatrix} x_1 x_2 end{bmatrix}
subject to begin{bmatrix} 1 & 1 F_1 & F_2 P_1 & P_2 end{bmatrix} begin{bmatrix} x_1 x_2 end{bmatrix} le begin{bmatrix} A F P end{bmatrix}, , begin{bmatrix} x_1 x_2 end{bmatrix} ge 0

Augmented form (slack form)

Linear programming problems must be converted into augmented form before being solved by the simplex algorithm. This form introduces non-negative slack variables to replace inequalities with equalities in the constraints. The problem can then be written in the following form:
Maximize Z in:
begin{bmatrix} 1 & -mathbf{c}^T & 0 0 & mathbf{A} & mathbf{I} end{bmatrix} begin{bmatrix} Z mathbf{x} mathbf{x}_s end{bmatrix} = begin{bmatrix} 0 mathbf{b} end{bmatrix}
mathbf{x}, , mathbf{x}_s ge 0
where mathbf{x}_s are the newly introduced slack variables, and Z is the variable to be maximized.

Example

The example above becomes as follows when converted into augmented form:
maximize S_1 x_1 + S_2 x_2, (objective function)
subject to x_1 + x_2 + x_3 = A, (augmented constraint)
F_1 x_1 + F_2 x_2 + x_4 = F, (augmented constraint)
P_1 x_1 + P_2 x_2 + x_5 = P, (augmented constraint)
x_1,x_2,x_3,x_4,x_5 ge 0
where x_3,x_4,x_5, are (non-negative) slack variables.

Which in matrix form becomes:

Maximize Z   in:
begin{bmatrix}
   1 & -S_1 & -S_2 & 0 & 0 & 0 
   0 &   1    &   1    & 1 & 0 & 0 
   0 &  F_1  &  F_2  & 0 & 1 & 0 
   0 &  P_1    & P_2 & 0 & 0 & 1 
end{bmatrix} begin{bmatrix}
   Z  x_1  x_2  x_3  x_4  x_5
end{bmatrix} = begin{bmatrix}
   0  A  F  P
end{bmatrix}, , begin{bmatrix}
   x_1  x_2  x_3  x_4  x_5
end{bmatrix} ge 0

Duality

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:
maximize mathbf{c}^T mathbf{x}
subject to mathbf{A}mathbf{x} le mathbf{b}, , mathbf{x} ge 0
The corresponding dual problem is:
minimize mathbf{b}^T mathbf{y}
subject to mathbf{A}^T mathbf{y} ge mathbf{c}, , mathbf{y} ge 0
where y is used instead of x as variable vector.

There are two ideas fundamental to duality theory. One is the fact that 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. The strong duality theorem states that if the primal has an optimal solution, x*, then the dual also has an optimal solution, y*, such that cTx*=bTy*.

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 also Farkas' lemma).

Example

Revisit the above example of the farmer who may grow wheat and barley with the set provision of some A land, F fertilizer and P insecticide. Assume now that 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), S1 for wheat and S2 for barley. This corresponds to the following linear programming problem:

minimize A y_A + F y_F + P y_P , (minimize the total cost of the means of production as the "objective function")
subject to y_A + F_1 y_F + P_1 y_P ge S_1 (the farmer must receive no less than S_1 for his wheat)
y_A + F_2 y_F + P_2 y_P ge S_2 (the farmer must receive no less than S_2 for his barley)
y_A ge 0,, y_F ge 0,, y_P ge 0 (prices cannot be negative)

Which in matrix form becomes:

minimize begin{bmatrix} A & F & P end{bmatrix} begin{bmatrix} y_A y_F y_P end{bmatrix}
subject to begin{bmatrix} 1 & F_1 & P_1 1 & F_2 & P_2 end{bmatrix} begin{bmatrix} y_A y_F y_P end{bmatrix} ge begin{bmatrix} S_1 S_2 end{bmatrix}, , begin{bmatrix} y_A y_F y_P end{bmatrix} ge 0

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.

The coefficients which 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 complementary slackness.

Complementary slackness

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:

Suppose that x = (x1, x2, . . ., xn) is primal feasible and that y = (y1, y2, . . . , ym) is dual feasible. Let (w1, w2, . . ., wm) denote the corresponding primal slack variables, and let (z1, z2, . . . , zn) denote the corresponding dual slack variables. Then x and y are optimal for their respective problems if and only if xjzj = 0, for j = 1, 2, . . . , n, wiyi = 0, for i = 1, 2, . . . , m.

So if the ith slack variable of the primal is not zero, then the ith variable of the dual is equal zero. Likewise, if the jth slack variable of the dual is not zero, then the jth variable of the primal is equal to zero.

Theory

Geometrically, the linear constraints define a convex polyhedron, which is called the feasible region. Since the objective function is also linear, hence a convex function, all local optima are automatically global optima (by the KKT theorem). The linearity of the objective function also implies that the set of optimal solutions is the convex hull of a finite set of points - usually a single point.

There are two situations in which no optimal solution can be found. First, if the constraints contradict each other (for instance, x ≥ 2 and x ≤ 1) then the feasible region is empty and there can be no optimal solution, since there are no solutions at all. In this case, the LP is said to be infeasible.

Alternatively, the polyhedron can be unbounded in the direction of the objective function (for example: maximize x1 + 3 x2 subject to x1 ≥ 0, x2 ≥ 0, x1 + x2 ≥ 10), in which case there is no optimal solution since solutions with arbitrarily high values of the objective function can be constructed.

Barring these two pathological conditions (which are often ruled out by resource constraints integral to the problem being represented, as above), the optimum is always attained at a vertex of the polyhedron. However, the optimum is not necessarily unique: it is possible to have a set of optimal solutions covering an edge or face of the polyhedron, or even the entire polyhedron (This last situation would occur if the objective function were constant).

Algorithms

The simplex algorithm, developed by George Dantzig, solves LP problems by constructing an admissible solution at a vertex of the polyhedron and then walking along edges of the polyhedron to vertices with successively higher values of the objective function until the optimum is reached. Although this algorithm is quite efficient in practice and can be guaranteed to find the global optimum if certain precautions against cycling are taken, it has poor worst-case behavior: it is possible to construct a linear programming problem for which the simplex method takes a number of steps exponential in the problem size. In fact, for some time it was not known whether the linear programming problem was solvable in polynomial time (complexity class P).

This long standing issue was resolved by Leonid Khachiyan in 1979 with the introduction of the ellipsoid method, the first worst-case polynomial-time algorithm for linear programming. To solve a problem which has n variables and can be encoded in L input bits, this algorithm uses O(n4L) arithmetic operations on numbers with O(L) digits. It consists of a specialization of the nonlinear optimization technique developed by Naum Z. Shor, generalizing the ellipsoid method for convex optimization proposed by Arkadi Nemirovski, a 2003 John von Neumann Theory Prize winner, and D. Yudin.

Khachiyan's algorithm was of landmark importance for establishing the polynomial-time solvability of linear programs. The algorithm had little practical impact, as the simplex method is more efficient for all but specially constructed families of linear programs. However, it inspired new lines of research in linear programming with the development of interior point methods, which can be implemented as a practical tool. In contrast to the simplex algorithm, which finds the optimal solution by progresses along points on the boundary of a polyhedral set, interior point methods move through the interior of the feasible region.

In 1984, N. Karmarkar proposed a new interior point projective method for linear programming. Karmarkar's algorithm not only improved on Khachiyan's theoretical worst-case polynomial bound (giving O(n^{3.5}L)), but also promised dramatic practical performance improvements over the simplex method. Since then, many interior point methods have been proposed and analyzed. Early successful implementations were based on affine scaling variants of the method. For both theoretical and practical properties, barrier function or path-following methods are the most common recently.

The current opinion is that the efficiency of good implementations of simplex-based methods and interior point methods is similar for routine applications of linear programming.

LP solvers are in widespread use for optimization of various problems in industry, such as optimization of flow in transportation networks, many of which can be transformed into linear programming problems only with some difficulty.

Open problems and recent work

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.

  • Does LP admit a strongly polynomial-time algorithm?
  • Does LP admit a strongly polynomial algorithm to find a strictly complementary solution?
  • Does LP admit a polynomial algorithm in the real number (unit cost) model of computation?

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." While algorithms exist to solve linear programming in weakly polynomial time, such as the ellipsoid methods and interior-point techniques, no algorithms have yet been found that allow strongly polynomial-time performance in the number of constraints and the number of variables. The development of such algorithms would be of great theoretical interest, and perhaps allow practical gains in solving large LPs as well.

  • Are there pivot rules which lead to polynomial-time Simplex variants?
  • Do all polyhedral graphs have polynomially-bounded diameter?
  • Is the Hirsch conjecture true for polyhedral graphs?

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 polyhedron. This means that their theoretical performance is limited by the maximum number of edges between any two vertices on the LP polyhedron. As a result, we are interested in knowing the maximum graph-theoretical diameter of polyhedral graphs. It has been proved that all polyhedra have subexponential diameter, and all experimentally observed polyhedra have linear diameter, it is presently unknown whether any polyhedron has superpolynomial or even superlinear diameter. If any such polyhedra exist, then no edge-following variant can run in polynomial or linear time, respectively. Questions about polyhedron 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. Pivot methods of this type have been studied since the 1970s. Essentially, these methods attempt to find the shortest pivot path on the arrangement polytope under the linear programming problem. In contrast to polyhedral graphs, graphs of arrangement polytopes have small diameter, allowing the possibility of strongly polynomial-time criss-cross pivot method without resolving questions about the diameter of general polyhedra.

Integer unknowns

If the unknown variables are all required to be integers, then the problem is called an integer programming (IP) or integer linear programming (ILP) problem. In contrast to linear programming, which can be solved efficiently in the worst case, integer programming problems are in many practical situations (those with bounded variables) NP-hard. 0-1 integer programming or binary integer programming (BIP) is the special case of integer programming where variables are required to be 0 or 1 (rather than arbitrary integers). This problem is also classified as NP-hard, and in fact the decision version was one of Karp's 21 NP-complete problems.

If only some of the unknown variables are required to be integers, then the problem is called a mixed integer programming (MIP) problem. These are generally also NP-hard.

There are however some important subclasses of IP and MIP problems that are efficiently solvable, most notably problems where the constraint matrix is totally unimodular and the right-hand sides of the constraints are integers.

Advanced algorithms for solving integer linear programs include:

Solvers and Scripting (Programming) Languages

See also

References

Further reading

  • Alexander Schrijver, Theory of Linear and Integer Programming. John Wiley & sons, 1998, ISBN 0-471-98232-6
  • Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapter 29: Linear Programming, pp.770–821.
  • Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5. A6: MP1: INTEGER PROGRAMMING, pg.245.
  • Bernd Gärtner, Jiri Matousek (2006). Understanding and Using Linear Programming, Berlin: Springer. ISBN 3-540-30697-8
  • V. Chandru and M.R.Rao, Linear Programming, Chapter 31 in Algorithms and Theory of Computation Handbook, edited by M.J.Atallah, CRC Press 1999, 31-1 to 31-37.
  • V. Chandru and M.R.Rao, Integer Programming, Chapter 32 in Algorithms and Theory of Computation Handbook, edited by M.J.Atallah, CRC Press 1999, 32-1 to 32-45.
  • Michael J. Todd (2002). "The many facets of linear programming". Mathematical Programming 91 (3):
  • Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf (2000). Computational Geometry. 2nd revised edition, Springer-Verlag. ISBN 3-540-65620-0. Chapter 4: Linear Programming: pp.63–94. Describes a randomized half-plane intersection algorithm for linear programming.
  • Jalaluddin Abdullah, Optimization by the Fixed-Point Method, Version 1.97.

External links

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