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Mathematical Solution Techniques

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Business Optimization Using Mathematical Programming

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 307))

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Abstract

This chapter provides some of the mathematical and algorithmic backgrounds to solving LP and MILP problems.

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Notes

  1. 1.

    Other standard notations are possible. One that is also used very often is subject to A x = b, l ≤x ≤u. Note that instead of the non-negativity constraints, we have lower and upper bounds on the variables. See Sect. 3.8.3 for how this case is treated.

  2. 2.

    We call a certain equation f i(x) = 0 of a system f(x) = 0 of equations redundant if it does not provide any further information with respect to the solution of that system. All information carried by f i(x) = 0 is already contained in the other equations.

  3. 3.

    To readers with a background in linear algebra, the concept of basic variables stems from linear algebra. Consider the constraint matrix A and a collection of m columns, A j. If these columns are linearly independent, then the corresponding variables x j are called basic variables. Since there may exist different sets of m linearly independent column vectors, we also have different sets of basic variables. Thus, to say a variable is a basic variable makes sense only if this variable is seen as a member of an ensemble of variables. The situation with canonical variables is different. It can easily be seen whether a variable is a canonical one or not. If we inspect the columns of A associated with a set of m canonical variables, we see that, after ordering the columns, we get the identity matrix, which is a very special example of a set of linearly independent columns.

  4. 4.

    In the original Simplex algorithm , the matrix A is updated in every iteration.

  5. 5.

    An inequality is said to be active for some values of the variables if the left- and right-hand sides are equal, i.e., for these values, the inequality is fulfilled as an equality. Example: x + y ≤ 5 becomes active for x = 3 and y = 2.

  6. 6.

    Such solvers are discussed in the review article by Freund & Mizuno (1996, [201]). Further details are found in Andersen & Ye (1995,[22]) and Vanderbei (1996,[560],2014,[561]).

  7. 7.

    Imagine the case that we have n binary variables, δ i, we have 2n possible combinations. While 23 = 8 and 210 = 1024, 250 is already a number that starts with leading digit 1 and then has 15 zeros. Note that we have approximately 2n ≈ 100.3n or to be precise 2n ≈ 10M⋅n with \(M=\ln (2)/\ln (10)\approx 0.30103\ldots \).

  8. 8.

    The problem size of a model may be expressed by the number of variables and constraints.

  9. 9.

    Tightening of bounds and other presolving operations are further described in Sect. 9.1.1.2.

  10. 10.

    Note that this property is not valid in MILP problems.

  11. 11.

    Using the results \({\mathbf {u}}^{\mathrm {T}}\mathbf {v}=\left ( {\mathbf {u}}^{ \mathrm {T}}\mathbf {v}\right ) ^{\mathrm {T}}={\mathbf {v}}^{\mathrm {T}}\mathbf {u}\) and \(\left ( {\mathbf {y}}^{\mathrm {T}}\mathsf {A}\right ) ^{\mathrm {T}}=\mathsf {A} ^{\mathrm {T}}\mathbf {y}\) from linear algebra, it is easy to see that both formulations of the dual problem are equivalent.

  12. 12.

    This result is only true if no upper bounds (see Sect. 3.8.3) are present.

  13. 13.

    As further pointed out on page 40, the basis inverse is only rarely inverted explicitly.

  14. 14.

    From now on A j denotes the columns of the original matrix A corresponding to the variable x j.

  15. 15.

    In this case, we are solving a maximization problem.

  16. 16.

    The “Boat Renting” problem shows this property very well. Let us inspect the system of linear equations in each iteration. The columns associated with the variables s 1, …, s 4, the original basic variables, give the basis inverse associated with the current basis. The reason is that elementary row operations are equivalent to a multiplication of the matrix representing the equations by another matrix, say M. If we inspect the first iteration, we can understand how the method works. The initial matrix and, in particular, the columns corresponding to the new basic variables are multiplied by M and obviously give , where is the unit matrix. Thus we have \( \mathsf {M}=\mathcal {B}^{-1}\). Since we have multiplied all columns of A by M, and in particular also the unit matrix associated with the original basic variables, these columns just give the columns of the basis inverse \(\mathcal {B}^{-1}\). In each iteration k we multiply our original matrix by such a matrix M k, so the original basic columns represent the product of all matrices M k, which then is the basis inverse of the current matrix.

  17. 17.

    Note that LP solvers may have a different sign convention for the shadow prices and may print out π 2 = π 3 = +0.5.

  18. 18.

    Affine scaling methods are based on a series of strictly feasible points (see definition below) of the primal problem. Around these points the feasible region of the primal problem is locally approximated by the so-called Dikin ellipsoid. This procedure leads to a convex optimization problem which is in most parts linear except for one convex quadratic constraint.

  19. 19.

    Potential reduction methods are those that have been introduced in Karmarkar’s famous 1984 paper. The objective function is the potential function \(q\ln \left ( {\mathbf {c}}^{\mathrm {T}}\mathbf {x}-{\mathbf {b}}^{\mathrm {T }}\mathbf {y}\right ) -\sum _{j=1}^{n}\ln x_{j}\) built up by the logarithm of the duality gap and a logarithmic term designed to repel feasible points from the boundary of the feasible region. The constraints are the linear constraints for primal and dual feasibility.

  20. 20.

    Central trajectory methods are primal–dual methods based on the concept of the central path or central trajectory . They are discussed in detail below. In their primal–dual predictor–corrector version, they are the most efficient ones, and according to Freund & Mizuno (1996), they have the most aesthetic qualities.

  21. 21.

    The reason for using this expression is that the method will include both the primal and dual variables.

  22. 22.

    diag(x 1, …, x n) denotes a diagonal matrix with diagonal elements x 1, …, x n and zeros at all other entries.

  23. 23.

    The screen output of an interior-point solver may thus contain the number of the current iteration, quantities measuring the violation of primal and dual feasibility, the values of the primal and the dual objective function (or the duality gap) and possibly the barrier parameter as valuable information on each iteration.

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  1. Department of Astronomy, University of Florida, Gainesville, FL, USA

    Josef Kallrath

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Kallrath, J. (2021). Mathematical Solution Techniques. In: Business Optimization Using Mathematical Programming. International Series in Operations Research & Management Science, vol 307. Springer, Cham. https://doi.org/10.1007/978-3-030-73237-0_3

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