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In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. It was first introduced by charnes and cooper in 1959 and further developed by miller and wagner in 1965 Lagrange multiplier in mathematical optimization, the method of lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables)

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Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives Chance constrained programming (ccp) is a mathematical optimization approach used to handle problems under uncertainty [1][2] it is generally divided into two subfields

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Discrete optimization and continuous optimization.

The use of optimization software requires that the function f is defined in a suitable programming language and connected at compilation or run time to the optimization software. Quadratic programming (qp) is the process of solving certain mathematical optimization problems involving quadratic functions Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constraints on the variables Quadratic programming is a type of nonlinear programming.

More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Structural optimization problems consist of two levels of optimization task and are commonly referred as mathematical programming problems with equilibrium constraints (mpec) The upper level objective in such problems may involve cost minimization or weight minimization subject to bounds on displacements, stresses and contact forces.

In a discrete optimization problem, such as combinatorial optimization, the search space could consist of a finite set of permutations, combinations, or configurations.