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3 Examples. 4 Geometric interpretation. ... In the theory of linear programming, a basic feasible solution (BFS) is a solution with a minimal set of non-zero variables.
For example, if the feasible region is defined by the constraint set {x ≥ 0, y ≥ 0}, then the problem of maximizing x + y has no optimum since any candidate solution can be improved upon by increasing x or y; yet if the problem is to minimize x + y, then there is an optimum (specifically at (x, y) = (0, 0)).
However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere).
A. The feasible set {b+L} ∩ K is bounded, and intersects the interior of the cone K. B. We are given in advance a strictly-feasible solution x^, that is, a feasible solution in the interior of K. C. We know in advance the optimal objective value, c*, of the problem. D. We are given an M-logarithmically-homogeneous self-concordant barrier F ...
feasible problem is one for which there exists at least one set of values for the choice variables satisfying all the constraints. an infeasible problem is one for which no set of values for the choice variables satisfies all the constraints. That is, the constraints are mutually contradictory, and no solution exists; the feasible set is the ...
The weak duality theorem says that, for each feasible solution x of the primal and each feasible solution y of the dual: c T x ≤ b T y. In other words, the objective value in each feasible solution of the dual is an upper-bound on the objective value of the primal, and objective value in each feasible solution of the primal is a lower-bound ...
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". This problem might have ...
A feasible solution that minimizes (or maximizes) the objective function is called an optimal solution. In mathematics, conventional optimization problems are usually stated in terms of minimization. A local minimum x* is defined as an element for which there exists some δ > 0 such that