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constraint programming These branches are all closely intertwined however, since many combinatorial optimization problems can be modeled as integer programs (e.g. shortest path ) or constraint programs, any constraint program can be formulated as an integer program and vice versa, and constraint and integer programs can often be given a ...
If the objective function and all of the hard constraints are linear and some hard constraints are inequalities, then the problem is a linear programming problem. This can be solved by the simplex method , which usually works in polynomial time in the problem size but is not guaranteed to, or by interior point methods which are guaranteed to ...
Constraint programming is an embedding of constraints in a host language. The first host languages used were logic programming languages, so the field was initially called constraint logic programming. The two paradigms share many important features, like logical variables and backtracking.
To see this, note that the two constraints x 1 (x 1 − 1) ≤ 0 and x 1 (x 1 − 1) ≥ 0 are equivalent to the constraint x 1 (x 1 − 1) = 0, which is in turn equivalent to the constraint x 1 ∈ {0, 1}. Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained ...
The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function (+, +) to lie in the second-order cone in +. [ 1 ] SOCPs can be solved by interior point methods [ 2 ] and in general, can be solved more efficiently than semidefinite programming (SDP) problems. [ 3 ]
If an inequality constraint holds with equality at the optimal point, the constraint is said to be binding, as the point cannot be varied in the direction of the constraint even though doing so would improve the value of the objective function. If an inequality constraint holds as a strict inequality at the optimal point (that is, does not hold ...
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. Such a constraint set is called a polyhedron or a polytope if it is bounded.
Constraint programming originated from a formalization of equalities of terms in Prolog II, leading to a general framework for embedding constraints into a logic programming language. The most common host languages are Prolog , C++ , and Java , but other languages have been used as well.