Any linear optimization model will have decision variables, a linear or quadratic objective function, and linear constraints and bounds on the values of the decision variables. A mixed-integer optimization model will require some or all of the decision variables to take integer values. The model may require the objective function to be maximized or minimized whilst satisfying the constraints and bounds. By default, HiGHS minimizes the objective function.
Bounds and the objective function
The bounds on a decision variable are the least and greatest values that it may take, and infinite bounds can be specified. A linear objective function is given by a set of coefficients, one for each decision variable, and its value is the sum of products of coefficients and values of decision variables. The objective coefficients are often referred to as costs, and some may be zero. When a model has been solved, the optimal values of the decision variables are referred to as the (primal) solution.
Constraints and the feasible region
Linear constraints require linear functions of decision variables to lie between bounds, and infinite bounds can be specified. If the bounds are equal, then the constraint is an equation. If the bounds are both finite, then the constraint is said to be boxed or two-sided. The set of points satisfying linear constraints and bounds is known as the feasible region. Geometrically, this is a multi-dimensional convex polyhedron, whose extreme points are referred to as vertices.
The constraint matrix
The coefficients of the linear constraints are naturally viewed as rows of a matrix. The constraint coefficients associated with a particular decision variable form a column of the constraint matrix. Hence constraints are sometimes referred to as rows, and decision variables as columns. Constraint matrix coefficients may be zero. Indeed, for large practical models it is typical for most of the coefficients to be zero. When this property can be exploited to computational advantage, the matrix is said to be sparse. When the constraint matrix is not sparse, the solution of large models is normally intractable computationally.
It is possible to define a set of constraints and bounds that cannot be satisfied, in which case the model is said to be infeasible. Conversely, it is possible that the value of the objective function can be improved without bound whilst satisfying the constraints and bounds, in which case the model is said to be unbounded. If a model is neither infeasible, nor unbounded, it has an optimal solution. The optimal objective function value for a linear optimization model may be achieved at more than point, in which case the optimal solution is said to be non-unique.
The values of the decision variables are referred to as primal values to distingush them from dual values.
When none of the decision variables is required to take integer values, the model is said to be continuous. For continuous models, each variable and constraint has an associated dual variable. The values of the dual variables constitute the dual solution, and it is for this reason that the term primal solution is used to distinguish the optimal values of the decision variables. At the optimal solution of a continuous model, some of the decision variables and values of constraint functions will be equal to their lower or upper bounds. Such a bound is said to be active. If a variable or constraint is at a bound, its corresponding dual solution value will generally be non-zero: when at a lower bound the dual value will be non-negative; when at an upper bound the dual value will be non-positive. When maximizing the objective the required signs of the dual values are reversed. Due to their economic interpretation, the dual values associated with constraints are often referred to as shadow prices or fair prices. Mathematically, the dual values associated with variables are often referred to as reduced costs, and the dual values associated with constraints are often referred to as Lagrange multipliers.
An LP model that is neither infeasible, nor unbounded, has an optimal solution at a vertex. At a vertex, the decision variables can be partitioned into as many basic variables as there are constraints, and nonbasic variables. Such a solution is known as a basic solution, and the partition referred to as a basis.
Analysis of the change in optimal objective value of a continuous linear optimization model as the cost coefficients and bounds are changed is referred to in HiGHS as ranging. For an active bound, the corresponding dual value gives the change in the objective if that bound is increased or decreased. This level of analysis is often referred to as sensitivity. In general, the change in the objective is only known for a limited range of values for the active bound. HiGHS will return the limits of these bound ranges ranges, the objective value at both limits and the index of a variable or constraint that will acquire an active bound at both limits. For each variable with an active bound, the solution will remain optimal for a range of values of its cost coefficient. HiGHS will return the values of these cost ranges. For a variable or constraint whose value is not at a bound, HiGHS will return the range of values that the variable or constraint can take, the objective values at the limits of the range, and the index of a variable or constraint with a bound that will become in active at both limits.
When solving a MIP, some or all the variables must take discrete values. In HiGHS there are three types of discrete variables.
- Integer: those that must take integer values between their bounds
- Semi-continuous: those that must be zero or take continuous values between their bounds
- Semi-integer: those that must be zero or take integer values between their bounds
In the following discussion, for ease of reference to relative objective values, it is assumed that the objective is being minimized
Any point for which the discrete variables satisfy their requirements, is said to be integer feasible. The objective value at such a point is an upper bound on the optimal objective value. The least such bound is known as the primal bound. The MIP solver generates a sequence of LPs, each of which has bounds on the variables that are tighter than those of the original model. When unsolved, there is a bound on the optimal objective value for each such LP and, the greatest such bound is known as the dual bound. The optimal objective value of the MIP cannot be less than the dual bound. Hence the gap between the primal and dual bounds is a measure of progress of the MIP solver. Although the absolute gap is of some interest, the gap relative to the primal bound is a better measure. When the gap reaches zero then the MIP is solved to optimality. However, it is often preferable to stop the MIP solver when the relative gap is below a specified tolerance.