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The function f is variously called an objective function, criterion function, loss function, cost function (minimization), [8] utility function or fitness function (maximization), or, in certain fields, an energy function or energy functional. A feasible solution that minimizes (or maximizes) the objective function is called an optimal solution.
In LP, the objective and constraint functions are all linear. Quadratic programming are the next-simplest. In QP, the constraints are all linear, but the objective may be a convex quadratic function. Second order cone programming are more general. Semidefinite programming are more general. Conic optimization are even more general - see figure ...
Convex feasible sets arise in many types of problems, including linear programming problems, and they are of particular interest because, if the problem has a convex objective function that is to be minimized, it will generally be easier to solve in the presence of a convex feasible set and any local optimum will also be a global optimum.
In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. [4]
The red line is a level set of the cost function, and the arrow indicates the direction in which we are optimizing. A closed feasible region of a problem with three variables is a convex polyhedron. The surfaces giving a fixed value of the objective function are planes (not shown). The linear programming problem is to find a point on the ...
In this example, the first line defines the function to be minimized (called the objective function, loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint.
If the objective function is concave (maximization problem), or convex (minimization problem) and the constraint set is convex, then the program is called convex and general methods from convex optimization can be used in most cases. If the objective function is quadratic and the constraints are linear, quadratic programming techniques are used.
In LP the objective function is a linear function, while the objective function of a linear–fractional program is a ratio of two linear functions. In other words, a linear program is a fractional–linear program in which the denominator is the constant function having the value one everywhere.