Optimal means the best under given conditions. Quality is assessed using an optimality criterion, and conditions are specified as constraints on additional criteria.
The drive to improve efficiency in labor, creative work, and any goal-driven activity is fundamentally expressed in the concept of optimality. The difference between the scientific and everyday understandings of optimality is minimal. Although phrases like "most optimal" or "maximum effect at minimum cost" are mathematically imprecise, such expressions loosely convey the intended idea. When specific optimization is required, people typically reformulate their statements accurately.
Optimization — in mathematics, computer science, and operations research — is the problem of finding an extremum (minimum or maximum) of an objective function in some region of a finite-dimensional vector space, bounded by a set of linear and/or nonlinear equalities and/or inequalities.
Optimization models are designed to determine the optimal (best) parameters of a modeled object with respect to some criterion, or to find the optimal (best) mode of controlling a process. Some of the model's parameters are classified as control parameters; by varying them, one can obtain different sets of output parameter values. As a rule, such models are constructed using one or more descriptive models and include a criterion that allows comparing different sets of output parameter values to select the best one. Constraints in the form of equalities and inequalities, related to the characteristics of the object or process under consideration, may be imposed on the domain of input parameter values. The purpose of optimization models is to find such admissible control parameters at which the selection criterion reaches its "best value."
The problem is formulated as a mathematical model. The standard mathematical model of operations research is presented in the following formulation:
Maximization or minimization of the objective function, subject to the fulfillment of constraints.
Solutions are called optimal if, by one attribute or another, they are preferable to others. Every choice of the best alternative is specific, since it is based on conformity to established criteria. When speaking of an optimal alternative, one specifies these criteria ("optimal with respect to…"). What is optimal under one criterion is not necessarily so under another.
A feasible solution is one that satisfies all the constraints of the model. In certain cases, the number of feasible solutions may be infinite.
An optimal solution is one that, in addition to being feasible, maximizes or minimizes the objective function.
Optimization is the maximization or minimization of the objective function.
An optimization model is a decision-making model that contains a measure of effectiveness (an objective function) to be optimized, subject to given constraints.
An optimal solution is an admissible set of values of the decision variables that optimizes the objective function of the optimization model.
A large number of practical choice problems reduce to finding the best or most preferred alternatives, and frequently to finding the single best alternative. In this process, each decision-maker has their own subjective notions of what is preferable for them in a specific choice situation.
Many problems allow for a mathematical choice model, where the best alternative is defined by one or more numerical effectiveness (or quality) criteria. These criteria, determined by the problem, are objective functions of variables representing properties of the alternatives. The decision-maker's most preferred alternative is then the optimal choice, yielding extremal values of effectiveness measures under given conditions.
A fundamental aspect of formulating the optimal choice problem is the ability to describe the problematic situation and the decision-maker's preferences in quantitative form. This means that, first, the possible solution alternatives (alternatives, objects, courses of action) are determined by quantitative attributes (variables, parameters, attributes) measured using numerical scales. Second, quantitative measures must be specified (optimality criteria, effectiveness measures, objective functions, value functions), by whose magnitude the quality of the chosen alternative is assessed. Such situations are characteristic of well-structured problems and recurring choice situations typical of operations research and optimal control.
To analyze possible alternatives for solving a problem (ways of achieving a goal) and to select the best alternative(s) among them, formal models of optimal choice are constructed. A model provides a simplified representation of the real problem and must reflect the most important, objectively existing dependencies and relationships among the alternatives, their descriptive attributes, and constraints imposed by controllable and uncontrollable factors. Constructing such a model is the task of consultant-analysts and experts, with the decision-maker's participation. When constructing a choice model, one must balance the adequacy and detail of the model against the precision required for the real choice problem, as well as against the volume of information needed to find the solution — both the information already available and that which can be additionally obtained.
Optimization problems are strictly formal mathematical problems. The practical significance of the solutions to such problems depends directly on the quality of the initial mathematical model. In complex systems, mathematical modeling is difficult, approximate, and imprecise. The more complex the system, the more cautiously one should approach its optimization.
From the standpoint of systems analysis, the attitude toward optimization can be formulated as follows: it is a powerful means of improving efficiency, but it should be used with increasing caution as problem complexity increases.
For all the obvious usefulness of the idea of optimization, practice demands that it be handled with care. There are sufficiently compelling grounds for such a conclusion.
- An optimal solution often proves unstable: seemingly insignificant changes in the problem's conditions can lead to the selection of substantially different alternatives.
- The system under consideration is part of some larger system, and local optimization does not necessarily yield the same result as would be required for the subsystem when optimizing the system as a whole. This necessitates aligning subsystem criteria with system criteria, often rendering local optimization unnecessary.
- Criteria characterize the goal only indirectly — sometimes better, sometimes worse, but always approximately. Maximization of the optimality criterion is often equated with the goal, but in reality these are different things. In effect, the criterion and the goal relate to each other as a model and the original, with all the attendant implications. Many goals are difficult or even impossible to describe quantitatively.
- Without specifying all necessary constraints, we may, simultaneously with maximizing the primary criterion, obtain unforeseen and undesirable side effects.