![]() Second-order cone programming (SOCP) is a convex program, and includes certain types of quadratic programs.Such a constraint set is called a polyhedron or a polytope if it is bounded. Linear programming (LP), a type of convex programming, studies the case in which the objective function f is linear and the constraints are specified using only linear equalities and inequalities.This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming. ![]() Convex programming studies the case when the objective function is convex (minimization) or concave (maximization) and the constraint set is convex. ![]() Other notable researchers in mathematical optimization include the following: ( Programming in this context does not refer to computer programming, but comes from the use of program by the United States military to refer to proposed training and logistics schedules, which were the problems Dantzig studied at that time.) Dantzig published the Simplex algorithm in 1947, and John von Neumann developed the theory of duality in the same year. Dantzig, although much of the theory had been introduced by Leonid Kantorovich in 1939. The term " linear programming" for certain optimization cases was due to George B. Operators arg min and arg max are sometimes also written as argmin and argmax, and stand for argument of the minimum and argument of the maximum.įermat and Lagrange found calculus-based formulae for identifying optima, while Newton and Gauss proposed iterative methods for moving towards an optimum. Many real-world and theoretical problems may be modeled in this general framework.į ( x 0 ) ≥ f ( x ) ⇔ − f ( x 0 ) ≤ − f ( x ), , where k ranges over all integers. Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, but still in use for example in linear programming – see History below). Given: a function f : A → ℝ from some set A to the real numbers Sought: an element x 0 ∈ A such that f( x 0) ≤ f( x) for all x ∈ A ("minimization") or such that f( x 0) ≥ f( x) for all x ∈ A ("maximization"). They can include constrained problems and multimodal problems.Īn optimization problem can be represented in the following way:
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