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Linear Programming Algorithms

Linear Programming Definition

Linear programming is the problem of finding a vector x that minimizes a linear function f^Tx subject to linear constraints:

$\min_{x} f^{T} x$

such that one or more of the following hold:

A·x ≤ b
Aeq·x = beq
l ≤ x ≤ u.

Interior-Point `linprog` Algorithm

The linprog 'interior-point' algorithm is very similar to the interior-point-convex quadprog Algorithm. It also shares many features with the linprog 'interior-point-legacy' algorithm. These algorithms have the same general outline:

Presolve, meaning simplification and conversion of the problem to a standard form.
Generate an initial point. The choice of an initial point is especially important for solving interior-point algorithms efficiently, and this step can be time-consuming.
Predictor-corrector iterations to solve the KKT equations. This step is generally the most time-consuming.

Presolve

The algorithm first tries to simplify the problem by removing redundancies and simplifying constraints. The tasks performed during the presolve step can include the following:

Check if any variables have equal upper and lower bounds. If so, check for feasibility, and then fix and remove the variables.
Check if any linear inequality constraint involves only one variable. If so, check for feasibility, and then change the linear constraint to a bound.
Check if any linear equality constraint involves only one variable. If so, check for feasibility, and then fix and remove the variable.
Check if any linear constraint matrix has zero rows. If so, check for feasibility, and then delete the rows.
Determine if the bounds and linear constraints are consistent.
Check if any variables appear only as linear terms in the objective function and do not appear in any linear constraint. If so, check for feasibility and boundedness, and then fix the variables at their appropriate bounds.
Change any linear inequality constraints to linear equality constraints by adding slack variables.

If the algorithm detects an infeasible or unbounded problem, it halts and issues an appropriate exit message.

The algorithm might arrive at a single feasible point, which represents the solution.

If the algorithm does not detect an infeasible or unbounded problem in the presolve step, and if the presolve has not produced the solution, the algorithm continues to its next steps. After reaching a stopping criterion, the algorithm reconstructs the original problem, undoing any presolve transformations. This final step is the postsolve step.

For simplicity, if the problem is not solved in the presolve step, the algorithm shifts all finite lower bounds to zero.

Generate Initial Point

To set the initial point, x0, the algorithm does the following.

Initialize x0 to ones(n,1), where n is the number of elements of the objective function vector f.
Convert all bounded components to have a lower bound of 0. If component i has a finite upper bound u(i), then x0(i) = u/2.
For components that have only one bound, modify the component if necessary to lie strictly inside the bound.
To put x0 close to the central path, take one predictor-corrector step, and then modify the resulting position and slack variables to lie well within any bounds. For details of the central path, see Nocedal and Wright [7], page 397.

Predictor-Corrector

Similar to the fmincon interior-point algorithm, the interior-point algorithm tries to find a point where the Karush-Kuhn-Tucker (KKT) conditions hold. To describe these equations for the linear programming problem, consider the standard form of the linear programming problem after preprocessing:

$\min_{x} f^{T} x subject to {\begin{matrix} \bar{A} x = \bar{b} \\ x + t = u \\ x, t \geq 0. \end{matrix}$

Assume for now that all variables have at least one finite bound. By shifting and negating components, if necessary, this assumption means that all x components have a lower bound of 0.
$\bar{A}$ is the extended linear matrix that includes both linear inequalities and linear equalities. $\bar{b}$ is the corresponding linear equality vector. $\bar{A}$ also includes terms for extending the vector x with slack variables s that turn inequality constraints to equality constraints:
$\bar{A} x = (\begin{matrix} A_{e q} & 0 \\ A & I \end{matrix}) (\begin{matrix} x_{0} \\ s \end{matrix}),$
where x₀ means the original x vector.
t is the vector of slacks that convert upper bounds to equalities.

The Lagrangian for this system involves the following vectors:

y, Lagrange multipliers associated with the linear equalities
v, Lagrange multipliers associated with the lower bound (positivity constraint)
w, Lagrange multipliers associated with the upper bound

The Lagrangian is

$L = f^{T} x - y^{T} (\bar{A} x - \bar{b}) - v^{T} x - w^{T} (u - x - t) .$

Therefore, the KKT conditions for this system are

$\begin{matrix} f - {\bar{A}}^{T} y - v + w = 0 \\ \bar{A} x = \bar{b} \\ x + t = u \\ v_{i} x_{i} = 0 \\ w_{i} t_{i} = 0 \\ (x, v, w, t) \geq 0. \end{matrix}$

The linprog algorithm uses a different sign convention for the returned Lagrange multipliers than this discussion gives. This discussion uses the same sign as most literature. See lambda.

The algorithm first predicts a step from the Newton-Raphson formula, and then computes a corrector step. The corrector attempts to reduce the residual in the nonlinear complementarity equations s_iz_i = 0. The Newton-Raphson step is

(\begin{matrix} 0 & - {\bar{A}}^{T} & 0 & - I & I \\ \bar{A} & 0 & 0 & 0 & 0 \\ - I & 0 & - I & 0 & 0 \\ V & 0 & 0 & X & 0 \\ 0 & 0 & W & 0 & T \end{matrix}) (\begin{matrix} Δ x \\ Δ y \\ Δ t \\ \begin{array}{l} Δ v \\ Δ w \end{array} \end{matrix}) = - (\begin{matrix} f - {\bar{A}}^{T} y - v + w \\ \bar{A} x - \bar{b} \\ u - x - t \\ \begin{array}{l} V X \\ W T \end{array} \end{matrix}) = - (\begin{matrix} r_{d} \\ r_{p} \\ r_{u b} \\ \begin{array}{l} r_{v x} \\ r_{w t} \end{array} \end{matrix}),

(1)

where X, V, W, and T are diagonal matrices corresponding to the vectors x, v, w, and t respectively. The residual vectors on the far right side of the equation are:

r_d, the dual residual
r_p, the primal residual
r_ub, the upper bound residual
r_vx, the lower bound complementarity residual
r_wt, the upper bound complementarity residual

The iterative display reports these quantities:

$\begin{array}{l} Primal infeasibility = {‖ r_{p} ‖}_{1} + {‖ r_{u b} ‖}_{1} \\ Dual infeasibility = {‖ r_{d} ‖}_{\infty} . \end{array}$

To solve Equation 1, first convert it to the symmetric matrix form

(\begin{matrix} - D & {\bar{A}}^{T} \\ \bar{A} & 0 \end{matrix}) (\begin{matrix} Δ x \\ Δ y \end{matrix}) = - (\begin{matrix} R \\ r_{p} \end{matrix}),

(2)

where

$\begin{array}{l} D = X^{- 1} V + T^{- 1} W \\ R = - r_{d} - X^{- 1} r_{v x} + T^{- 1} r_{w t} + T^{- 1} W r_{u b} . \end{array}$

All the matrix inverses in the definitions of D and R are simple to compute because the matrices are diagonal.

To derive Equation 2 from Equation 1, notice that the second row of Equation 2 is the same as the second matrix row of Equation 1. The first row of Equation 2 comes from solving the last two rows of Equation 1 for Δv and Δw, and then solving for Δt.

Equation 2 is symmetric, but it is not positive definite because of the –D term. Therefore, you cannot solve it using a Cholesky factorization. A few more steps lead to a different equation that is positive definite, and hence can be solved efficiently by Cholesky factorization.

The second set of rows of Equation 2 is

$\bar{A} Δ x = - r_{p}$

and the first set of rows is

$- D Δ x + {\bar{A}}^{T} Δ y = - R .$

Substituting

$Δ x = D^{- 1} {\bar{A}}^{T} Δ y + D^{- 1} R$

gives

\bar{A} D^{- 1} {\bar{A}}^{T} Δ y = - \bar{A} D^{- 1} R - r_{p} .

(3)

Usually, the most efficient way to find the Newton step is to solve Equation 3 for Δy using Cholesky factorization. Cholesky factorization is possible because the matrix multiplying Δy is obviously symmetric and, in the absence of degeneracies, is positive definite. Afterward, to find the Newton step, back substitute to find Δx, Δt, Δv, and Δw. However, when $\bar{A}$ has a dense column, it can be more efficient to solve Equation 2 instead. The linprog interior-point algorithm chooses the solution algorithm based on the density of columns.

For more algorithm details, see Mehrotra [6].

After calculating the corrected Newton step, the algorithm performs more calculations to get both a longer current step, and to prepare for better subsequent steps. These multiple correction calculations can improve both performance and robustness. For details, see Gondzio [4].

The predictor-corrector algorithm is largely the same as the full quadprog 'interior-point-convex' version, except for the quadratic terms. See Full Predictor-Corrector.

Stopping Conditions

The predictor-corrector algorithm iterates until it reaches a point that is feasible (satisfies the constraints to within tolerances) and where the relative step sizes are small. Specifically, define

$ρ = \max (1, ‖ \bar{A} ‖, ‖ f ‖, ‖ \bar{b} ‖) .$

The algorithm stops when all of these conditions are satisfied:

$\begin{matrix} {‖ r_{p} ‖}_{1} + {‖ r_{u b} ‖}_{1} \leq ρ TolCon \\ {‖ r_{d} ‖}_{\infty} \leq ρ TolFun \\ r_{c} \leq TolFun, \end{matrix}$

where

$r_{c} = \max_{i} (\min (| x_{i} v_{i} |, | x_{i} |, | v_{i} |), \min (| t_{i} w_{i} |, | t_{i} |, | w_{i} |)) .$

r_c essentially measures the size of the complementarity residuals xv and tw, which are each vectors of zeros at a solution.

Interior-Point-Legacy Linear Programming

Introduction

The interior-point-legacy method is based on LIPSOL ([52]), which is a variant of Mehrotra's predictor-corrector algorithm ([47]), a primal-dual interior-point method.

Main Algorithm

The algorithm begins by applying a series of preprocessing steps (see Preprocessing). After preprocessing, the problem has the form

\min_{x} f^{T} x such that {\begin{matrix} A \cdot x = b \\ 0 \leq x \leq u . \end{matrix}

(4)

The upper bounds constraints are implicitly included in the constraint matrix A. With the addition of primal slack variables s, Equation 4 becomes

\min_{x} f^{T} x such that {\begin{matrix} A \cdot x = b \\ x + s = u \\ x \geq 0, s \geq 0. \end{matrix}

(5)

which is referred to as the primal problem: x consists of the primal variables and s consists of the primal slack variables. The dual problem is

\max b^{T} y - u^{T} w such that {\begin{matrix} A^{T} \cdot y - w + z = f \\ z \geq 0, w \geq 0, \end{matrix}

(6)

where y and w consist of the dual variables and z consists of the dual slacks. The optimality conditions for this linear program, i.e., the primal Equation 5 and the dual Equation 6, are

\begin{array}{l} F (x, y, z, s, w) = (\begin{matrix} A \cdot x - b \\ x + s - u \\ A^{T} \cdot y - w + z - f \\ x_{i} z_{i} \\ s_{i} w_{i} \end{matrix}) = 0, \\ x \geq 0, z \geq 0, s \geq 0, w \geq 0, \end{array}

(7)

where x_iz_i and s_iw_i denote component-wise multiplication.

The linprog algorithm uses a different sign convention for the returned Lagrange multipliers than this discussion gives. This discussion uses the same sign as most literature. See lambda.

The quadratic equations x_iz_i = 0 and s_iw_i = 0 are called the complementarity conditions for the linear program; the other (linear) equations are called the feasibility conditions. The quantity

x^Tz + s^Tw

is the duality gap, which measures the residual of the complementarity portion of F when (x,z,s,w) ≥ 0.

The algorithm is a primal-dual algorithm, meaning that both the primal and the dual programs are solved simultaneously. It can be considered a Newton-like method, applied to the linear-quadratic system F(x,y,z,s,w) = 0 in Equation 7, while at the same time keeping the iterates x, z, w, and s positive, thus the name interior-point method. (The iterates are in the strictly interior region represented by the inequality constraints in Equation 5.)

The algorithm is a variant of the predictor-corrector algorithm proposed by Mehrotra. Consider an iterate v = [x;y;z;s;w], where [x;z;s;w] > 0 First compute the so-called prediction direction

$Δ v_{p} = - {(F^{T} (v))}^{- 1} F (v),$

which is the Newton direction; then the so-called corrector direction

$Δ v_{c} = - {(F^{T} (v))}^{- 1} F (v + Δ v_{p}) - μ \hat{e},$

where μ > 0 is called the centering parameter and must be chosen carefully. $\hat{e}$ is a zero-one vector with the ones corresponding to the quadratic equations in F(v), i.e., the perturbations are only applied to the complementarity conditions, which are all quadratic, but not to the feasibility conditions, which are all linear. The two directions are combined with a step length parameter α > 0 and update v to obtain the new iterate v⁺:

$v^{+} = v + α (Δ v_{p} + Δ v_{c}),$

where the step length parameter α is chosen so that

v⁺ = [x⁺;y⁺;z⁺;s⁺;w⁺]

satisfies

[x⁺;z⁺;s⁺;w⁺] > 0.

In solving for the preceding predictor/corrector directions, the algorithm computes a (sparse) direct factorization on a modification of the Cholesky factors of A·A^T. If A has dense columns, it instead uses the Sherman-Morrison formula. If that solution is not adequate (the residual is too large), it performs an LDL factorization of an augmented system form of the step equations to find a solution. (See Example 4 — The Structure of D in the MATLAB^® ldl function reference page.)

The algorithm then loops until the iterates converge. The main stopping criteria is a standard one:

$\max (\frac{‖ r_{b} ‖}{\max (1, ‖ b ‖)}, \frac{‖ r_{f} ‖}{\max (1, ‖ f ‖)}, \frac{‖ r_{u} ‖}{\max (1, ‖ u ‖)}, \frac{| f^{T} x - b^{T} y + u^{T} w |}{\max (1, | f^{T} x |, | b^{T} y - u^{T} w |)}) \leq t o l,$

where

$\begin{matrix} r_{b} = A x - b \\ r_{f} = A^{T} y - w + z - f \\ r_{u} = {x} + s - u \end{matrix}$

are the primal residual, dual residual, and upper-bound feasibility respectively ({x} means those x with finite upper bounds), and

$f^{T} x - b^{T} y + u^{T} w$

is the difference between the primal and dual objective values, and tol is some tolerance. The sum in the stopping criteria measures the total relative errors in the optimality conditions in Equation 7.

The measure of primal infeasibility is ||r_b||, and the measure of dual infeasibility is ||r_f||, where the norm is the Euclidean norm.

Preprocessing

The algorithm first tries to simplify the problem by removing redundancies and simplifying constraints. The tasks performed during the presolve step can include the following:

Check if any variables have equal upper and lower bounds. If so, check for feasibility, and then fix and remove the variables.
Check if any linear inequality constraint involves only one variable. If so, check for feasibility, and then change the linear constraint to a bound.
Check if any linear equality constraint involves only one variable. If so, check for feasibility, and then fix and remove the variable.
Check if any linear constraint matrix has zero rows. If so, check for feasibility, and then delete the rows.
Determine if the bounds and linear constraints are consistent.
Check if any variables appear only as linear terms in the objective function and do not appear in any linear constraint. If so, check for feasibility and boundedness, and then fix the variables at their appropriate bounds.
Change any linear inequality constraints to linear equality constraints by adding slack variables.

If the algorithm detects an infeasible or unbounded problem, it halts and issues an appropriate exit message.

The algorithm might arrive at a single feasible point, which represents the solution.

For simplicity, the algorithm shifts all lower bounds to zero.

While these preprocessing steps can do much to speed up the iterative part of the algorithm, if the Lagrange multipliers are required, the preprocessing steps must be undone since the multipliers calculated during the algorithm are for the transformed problem, and not the original. Thus, if the multipliers are not requested, this transformation back is not computed, and might save some time computationally.

Dual-Simplex Algorithm

At a high level, the linprog 'dual-simplex' algorithm essentially performs a simplex algorithm on the dual problem.

The algorithm begins by preprocessing as described in Preprocessing. For details, see Andersen and Andersen [1] and Nocedal and Wright [7], Chapter 13. This preprocessing reduces the original linear programming problem to the form of Equation 4:

$\min_{x} f^{T} x such that {\begin{matrix} A \cdot x = b \\ 0 \leq x \leq u . \end{matrix}$

A and b are transformed versions of the original constraint matrices. This is the primal problem.

Primal feasibility can be defined in terms of the ⁺ function

$x^{+} = {\begin{array}{l} x & if x > 0 \\ 0 & if x \leq 0. \end{array}$

The measure of primal infeasibility is

$Primal infeasibility = \sqrt{{({(lb - x)}^{+})}^{2} + {({(x - ub)}^{+})}^{2} + {({(A x - b)}^{+})}^{2} + {| Aeq x - beq |}^{2}} .$

As explained in Equation 6, the dual problem is to find vectors y and w, and a slack variable vector z that solve

$\max b^{T} y - u^{T} w such that {\begin{matrix} A^{T} \cdot y - w + z = f \\ z \geq 0, w \geq 0. \end{matrix}$

The linprog algorithm uses a different sign convention for the returned Lagrange multipliers than this discussion gives. This discussion uses the same sign as most literature. See lambda.

The measure of dual infeasibility is

$Dual infeasibility = {‖ A^{T} y + z - w - f ‖}_{2} .$

It is well known (for example, see [7]) that any finite solution of the dual problem gives a solution to the primal problem, and any finite solution of the primal problem gives a solution of the dual problem. Furthermore, if either the primal or dual problem is unbounded, then the other problem is infeasible. And if either the primal or dual problem is infeasible, then the other problem is either infeasible or unbounded. Therefore, the two problems are equivalent in terms of obtaining a finite solution, if one exists. Because the primal and dual problems are mathematically equivalent, but the computational steps differ, it can be better to solve the primal problem by solving the dual problem.

To help alleviate degeneracy (see Nocedal and Wright [7], page 366), the dual simplex algorithm begins by perturbing the objective function.

Phase 1 of the dual simplex algorithm is to find a dual feasible point. The algorithm does this by solving an auxiliary linear programming problem.

Phase 1 Outline

In phase 1, the algorithm finds an initial basic feasible solution (see Basic and Nonbasic Variables for a definition) by solving an auxiliary piecewise linear programming problem. The objective function of the auxiliary problem is the linear penalty function $P = \sum_{j} P_{j} (x_{j})$ ,

where P_j(x_j) is defined by

$P_{j} (x_{j}) = {\begin{array}{l} x_{j} - u_{j} & if x_{j} > u_{j} \\ 0 & if l_{j} \leq x_{j} \leq u_{j} \\ l_{j} - x_{j} & if l_{j} > x_{j} . \end{array}$

P(x) measures how much a point x violates the lower and upper bound conditions. The auxiliary problem is

$\min_{x} \sum_{j} P_{j} subject to {\begin{matrix} A \cdot x \leq b \\ A e q \cdot x = b e q . \end{matrix}$

The original problem has a feasible basis point if and only if the auxiliary problem has minimum value 0.

The algorithm finds an initial point for the auxiliary problem by a heuristic method that adds slack and artificial variables as necessary. The algorithm then uses this initial point together with the simplex algorithm to solve the auxiliary problem. The solution is the initial point for phase 2 of the main algorithm.

During Phase 2, the solver repeatedly chooses an entering variable and a leaving variable. The algorithm chooses a leaving variable according to a technique suggested by Forrest and Goldfarb [3] called dual steepest-edge pricing. The algorithm chooses an entering variable using the variation of Harris’ ratio test suggested by Koberstein [5]. To help alleviate degeneracy, the algorithm can introduce additional perturbations during Phase 2.

Phase 2 Outline

The solver iterates, attempting to maintain dual feasibility while reducing primal infeasibility, until the solution to the reduced, perturbed problem is both primal feasible and dual feasible. The algorithm unwinds the perturbations that it introduced. If the solution (to the perturbed problem) is dual infeasible for the unperturbed (original) problem, then the solver restores dual feasibility using primal simplex or Phase 1 algorithms. Finally, the solver unwinds the preprocessing steps to return the solution to the original problem.

Basic and Nonbasic Variables

This section defines the terms basis, nonbasis, and basic feasible solutions for a linear programming problem. The definition assumes that the problem is given in the following standard form:

$\min_{x} f^{T} x such that {\begin{matrix} A \cdot x = b, \\ l b \leq x \leq u b . \end{matrix}$

(Note that A and b are not the matrix and vector defining the inequalities in the original problem.) Assume that A is an m-by-n matrix, of rank m < n, whose columns are {a₁, a₂, ..., a_n}. Suppose that ${a_{i_{1}}, a_{i_{2}}, ..., a_{i_{m}}}$ is a basis for the column space of A, with index set B = {i₁, i₂, ..., i_m}, and that N = {1, 2, ..., n}\B is the complement of B. The submatrix A_B is called a basis and the complementary submatrix A_N is called a nonbasis. The vector of basic variables is x_B and the vector of nonbasic variables is x_N. At each iteration in phase 2, the algorithm replaces one column of the current basis with a column of the nonbasis and updates the variables x_B and x_N accordingly.

If x is a solution to A·x = b and all the nonbasic variables in x_N are equal to either their lower or upper bounds, x is called a basic solution. If, in addition, the basic variables in x_B satisfy their lower and upper bounds, so that x is a feasible point, x is called a basic feasible solution.

References

[1] Andersen, E. D., and K. D. Andersen. Presolving in linear programming. Math. Programming 71, 1995, pp. 221–245.

[2] Applegate, D. L., R. E. Bixby, V. Chvátal and W. J. Cook, The Traveling Salesman Problem: A Computational Study, Princeton University Press, 2007.

[3] Forrest, J. J., and D. Goldfarb. Steepest-edge simplex algorithms for linear programming. Math. Programming 57, 1992, pp. 341–374.

[4] Gondzio, J. “Multiple centrality corrections in a primal dual method for linear programming.” Computational Optimization and Applications, Volume 6, Number 2, 1996, pp. 137–156. Available at https://www.maths.ed.ac.uk/~gondzio/software/correctors.ps.

[5] Koberstein, A. Progress in the dual simplex algorithm for solving large scale LP problems: techniques for a fast and stable implementation. Computational Optim. and Application 41, 2008, pp. 185–204.

[6] Mehrotra, S. “On the Implementation of a Primal-Dual Interior Point Method.” SIAM Journal on Optimization, Vol. 2, 1992, pp 575–601.

[7] Nocedal, J., and S. J. Wright. Numerical Optimization, Second Edition. Springer Series in Operations Research, Springer-Verlag, 2006.

Linear Programming Algorithms

Linear Programming Definition

Interior-Point linprog Algorithm

Presolve

Generate Initial Point

Predictor-Corrector

Stopping Conditions

Interior-Point-Legacy Linear Programming

Introduction

Main Algorithm

Preprocessing

Dual-Simplex Algorithm

Basic and Nonbasic Variables

References

Interior-Point `linprog` Algorithm