What can we say about the solution of dual LP if primal Maximisation LP is unbounded?
What can we say about the solution of dual LP if primal Maximisation LP is unbounded?
Weak duality In particular, if the primal is unbounded (from above) then the dual has no feasible solution, and if the dual is unbounded (from below) then the primal has no feasible solution.
What is a complementarity constraint?
Intuitively, a complementarity constraint is a way to model a constraint that is combinatorial in nature since, for example, the complementary conditions imply that either x or y must be 0 (both may be 0 as well). In this way, Knitro can recognize these constraints and handle them with special care internally.
Is it possible that both primal and dual are infeasible?
Primal and dual feasible and bounded is possible: Example is c = b = (0) and A = (0). Primal feasible and bounded, dual infeasible is impossible: If the primal has an optimal solution, the duality theorem tells us that the dual has an optimal solution as well. In particular the dual is feasible.
Why if Primal is maximization then Dual is minimization?
If the primal problem is a maximization problem, then the dual problem is a minimization problem and vice versa. All primal and dual variables must be non-negative (≥0). Types of Primal –Dual Problem. There are three types of Primal- Dual problems.
What are primal problems?
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 of the objective function subject to the constraints.
Why are slack variables non negative?
Slack variables are used in particular in linear programming. As with the other variables in the augmented constraints, the slack variable cannot take on negative values, as the simplex algorithm requires them to be positive or zero.
What are the complementary slackness conditions?
The condition that either (i) λ = 0 and g ( x *) ≤ c or (ii) λ ≥ 0 and g ( x *) = c is called a complementary slackness condition. For a problem with many constraints, then as before we introduce one multiplier for each constraint and obtain the Kuhn-Tucker conditions, defined as follows.
What does complementary slackness say about the Lagrangian multiplier?
If we think of the constraint as a resource constraint (e.g., materials, space, time, etc.) and the Lagrangian multiplier as the value or price of that resource, then Complementary Slackness is saying: If the constraint is binding, it means we are running out of this resource when maximizing the objective.
What is condition 2 in Lagrangian analysis?
Condition (2), usually referred as dual feasibility, states that x is also a feasible solution to the dual problem. λ and v are called the Lagrangian multipliers (or dual variables) corresponding to the constraints Ax ≥ b and x ≥ 0, respectively. Finally, condition (3) is called complementary slackness.
What are sufficient conditions for an optimal solution to NLP?
Necessary Conditions for an Optimal Solution. Condition (4.6c) is called the complementary slackness condition. The following theorem provides sufficient conditions for x* to be an optimal solution to the NLP given in (4.4). Theorem. Sufficient Conditions for an Optimal Solution. Suppose each g, (x) is a convex function.
