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To minimize the deviation, the problem is formulated in a basic form as:Īs the objective function, and linear constraints are , where i is the observation, gives the deviation, is an observation. In completing this algorithm, the optimal solution to the original problem will be Applications Minimizing the sum of absolute deviations Solve the following primal linear program to determine new dual optimal variables. If either condition occurs, then stop iterating. Or second, if the maximum number of iterations has been reached.

First, if there are zero components satisfying. īegin by solving the following linear program to determine initial primal and dual optimal solutions.Ĭheck to see if either of two conditions are true. Note that the algorithm starts at iteration. Typical values are approximately 10^-6, 10^-8, and 10, respectively. The algorithm is summarized as follows.Ĭhoose a value, a tolerance, and a maximum number of iterations. Olvi Mangasarian of the Computer Sciences Department of the University of Wisconsin-Madison worked on an algorithm for solving absolute value problems in this way, described in the paper Absolute Value Equation Solution via Linear Programming. Successive linear programming methods involve generating and solving a sequence of linear programming problems to ultimately solve one absolute value problem. In the former two cases, the model can be reformulated by introducing a new variable,, replacing in the original objective function with, and adding two extra constraints and. If the objective is a minimization problem of the form or is a maximization problem of the form, unfortunately, the model cannot be reformulated as a standard LP model, and instead must be solved using mixed-integer linear programming. If the objective is a minimization problem of the form or is a maximization problem of the form, then the model can easily be reformulated to be solved using linear programming. Absolute values in the objective functionĪbsolute values as part of the objective function of a model can also be reformulated to become linear, in certain cases. The same logic can be used to reformulate as and, or into and, for example.

This number line represents both the absolute value function as well as the two combined linear functions described above, demonstrating that the two formulations are equivalent. This relation is easiest to see using a number line, as follows:įigure 1: Number line depicting the above absolute value problem. In the case, the expression can be reformulated as and.

This methodology is the basis of performing linear programming with absolute values.

This function is effectively the combination two piecewise functions: if and if. In other words, can be reformulated into two linear expressions if the function is linear itself.įor the simplest case, take. If an absolute value function contains a linear function, then the absolute value can be reformulated into two linear expressions. However, through simple manipulation of the absolute value expression, these difficulties can be avoided and the problem can be solved using linear programming. They are not continuously differentiable functions, are nonlinear, and are relatively difficult to operate on. Optimization with absolute values is a special case of linear programming in which a problem made nonlinear due to the presence of absolute values is solved using linear programming methods.Ībsolute value functions themselves are very difficult to perform standard optimization procedures on. 3.2 Minimizing the maximum of absolute values.3.1 Minimizing the sum of absolute deviations.2.1.2 Absolute values in the objective function.

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