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The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal.
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, [1] whereas mathematical optimization is in general NP-hard. [2 ...
In matrix form, we can express the primal problem as: Maximize c T x subject to Ax ≤ b, x ≥ 0; with the corresponding symmetric dual problem, Minimize b T y subject to A T y ≥ c, y ≥ 0. An alternative primal formulation is: Maximize c T x subject to Ax ≤ b; with the corresponding asymmetric dual problem, Minimize b T y subject to A T ...
For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure m 0. For example, if there is a graph G which contains vertices u and v , an optimization problem might be "find a path from u to v that uses the fewest edges".
The knapsack problem is one of the most studied problems in combinatorial optimization, with many real-life applications. For this reason, many special cases and generalizations have been examined. For this reason, many special cases and generalizations have been examined.
Consider the following nonlinear optimization problem in standard form: . minimize () subject to (),() =where is the optimization variable chosen from a convex subset of , is the objective or utility function, (=, …,) are the inequality constraint functions and (=, …,) are the equality constraint functions.
Linear programming problems are optimization problems in which the objective function and the constraints are all linear. In the primal problem, the objective function is a linear combination of n variables. There are m constraints, each of which places an upper bound on a linear combination of the n variables. The goal is to maximize the value ...
After the problem on variables +, …, is solved, its optimal cost can be used as an upper bound while solving the other problems, In particular, the cost estimate of a solution having x i + 1 , … , x n {\displaystyle x_{i+1},\ldots ,x_{n}} as unassigned variables is added to the cost that derives from the evaluated variables.