optimiarvo

Optimiarvo is a term used in optimization theory to denote the value of the objective function at an optimal solution. It represents the best achievable outcome under a given set of constraints and applies to problems of both maximization and minimization. The word combines the Finnish roots optimi- (optimal) and arvo (value), and is sometimes used in Finnish-language texts to distinguish the value from the optimal solution itself.

Formally, consider a problem of maximizing or minimizing f(x) subject to x belonging to a feasible set X. The optimiarvo is f(x*) where x* is an optimal solution in X. If several optimal solutions exist, they all attain the same optimiarvo, so the value is well-defined even when the argmax or argmin is not unique.

In well-posed problems, the optimiarvo is unique; multiple optimal solutions may exist, but they share the same objective value. Nonconvexity or discrete decision spaces can produce multiple optimiarvo-achieving solutions, though each achieves the same optimal value.

Computing the optimiarvo typically involves solving the optimization problem to obtain x*, then evaluating f at that point. Algorithms may search for x* directly via gradient methods, linear or nonlinear programming, or combinatorial techniques, or estimate the optimiarvo through duality, relaxation, or surrogate models when exact solutions are intractable.

Examples include profit maximization in production planning, where the optimiarvo is the maximum attainable profit, and model training in machine learning, where the optimiarvo corresponds to the minimum value of the loss function. The concept is often distinguished from the optimal solution vector, which specifies the decision variables achieving the optimiarvo.