elements: . In that case, the objective can be written as: . Note that is increasing with when k<t, and decreasing with when k>t. Therefore, the maximum value is attained when k=t, that is, is larger than the smallest t objective values; in that case, the objective exactly equals , as claimed. Therefore, the problem (L2) is equivalent to the following lexicographic maximization problem: This problem (L4) has additional variables, and additional constraints. It can be solved by every algorithm for solving lexicographic maximization, for example: the sequential algorithm using n linear programs, or the lexicographic simplex algorithm (if the objectives and constraints are linear). Ordered Values Algorithm for general problems The Ordered Values Algorithm works in any domain in which the set of possible values of the objective functions is finite. It was developed by Ogryczak and Śliwiński. Let be the set of all values that can be returned by the functions , such that . Given a solution x, and an integer k in {1,..,r}, define as the number of occurences of the value vr in the vector . Then, the lexmaxmin problem can be stated as the following lexicographic minimization problem:since we want to have as few as possible functions attaining the smallest value; subject to this, as few as possible functions attaining the next-smallest value; and so on. Ogryczak and Śliwiński show how to transform this non-linear program into a linear program with auxiliary variables. In their computational experiments, the Ordered Values algorithm runs much faster than the Saturation algorithm and the Ordered Outcomes algorithm. Behringer's algorithm for quasiconcave functions Behringer presented a sequential algorithm for lexmaxmin optimization when the objectives are quasiconvex functions, and the feasible set X is a convex set. Weighted average Yager presented a way to represent the leximin ordering analytically using the Ordered weighted averaging aggregation operator. He assumes that all objective values are real numbers between 0 and 1, and the smallest difference between any two possible values is some constant d < 1 (so that values with difference smaller than d are considered equal). The weight of is set to approximately . This guarantees that maximizing the weighted sum is equivalent to lexmaxmin. Algorithms for discrete variables If the set of vectors is discrete, and the domain is sufficiently small, then it is possible to use one of the functions representing the leximin order, and maximize it subject to