# Primal dual

Click to print Opens in new window My article on linear programming is a prerequisite. For the second half dual simplex and degeneracymy article on duality in linear programming is a prerequisite. Linear programs have an amazing structure. This value is always greater than or equal to 0. The duality gap is zero if and only if strong duality holds. Otherwise the gap is strictly positive and weak duality holds. This alternative "duality gap" quantifies the discrepancy between the value of a current feasible but suboptimal iterate for the primal problem and the value of the dual problem; the value of the dual problem is, under regularity conditions, equal to the value of the convex relaxation of the primal problem: The convex relaxation is the problem arising replacing a non-convex feasible set with its closed convex hull and with replacing a non-convex function with its convex closurethat is the function that has the epigraph that is the closed convex hull of the original primal objective function.

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. A solution is a vector a list of n values that achieves the maximum Primal dual for the objective function.

In the dual problem, the objective function is a linear combination of the m values that are the limits in the m constraints from the primal problem. There are n dual constraints, each of which places a lower bound on a linear combination of m dual variables.

Relationship between the primal problem and the dual problem[ edit ] In the linear case, in the primal problem, from each sub-optimal point that satisfies all the constraints, there is a direction or subspace of directions to move that increases the objective function. Moving in any such direction is said to remove slack between the candidate solution and one or more constraints. An infeasible value of the candidate solution is one that exceeds one or more of the constraints. In the dual problem, the dual vector multiplies the constraints that determine the positions of the constraints in the primal.

Varying the dual vector in the dual problem is equivalent to revising the upper bounds in the primal problem. The lowest upper bound is sought.

That is, the dual vector is minimized in order to remove slack between the candidate positions of the constraints and the actual optimum.

An infeasible value of the dual vector is one that is too low. It sets the candidate positions of one or more of the constraints in a position that excludes the actual optimum. This intuition is made formal by the equations in Linear programming: Economic interpretation[ edit ] If we interpret our primal LP problem as a classical "resource allocation" problem, its dual can be interpreted as a " resource valuation " problem.

Nonlinear case[ edit ] In nonlinear programmingthe constraints are not necessarily linear. Nonetheless, many of the same principles apply. To ensure that the global maximum of a non-linear problem can be identified easily, the problem formulation often requires that the functions be convex and have compact lower level sets.

This is the significance of the Karush—Kuhn—Tucker conditions. They provide necessary conditions for identifying local optima of non-linear programming problems.

There are additional conditions constraint qualifications that are necessary so that it will be possible to define the direction to an optimal solution. An optimal solution is one that is a local optimum, but possibly not a global optimum.

The strong Lagrangian principle:and dual solutions, a dual variable can only be nonzero if the corresponding con-straint is tight in the primal, and a primal variable can only be nonzero if the corre-sponding constraint is tight in the dual.

Note that for a primal LP in standard form all primal constraints are equality constraints so any feasible point has all constraints tight. ComPLx: A Competitive Primal-dual Lagrange Optimization for Global Placement Myung-Chul Kim and Igor L. Markov University of Michigan, EECS Department, Ann Arbor, MI Relations between Primal and Dual If the primal problem is Maximize ctx subject to Ax = b, x ‚ 0 then the dual is Minimize bty subject to Aty ‚ c (and y unrestricted) Easy fact: If x is feasible for the primal, and y is feasible for the dual, then ctx • bty So (primal optimal) • (dual optimal) (Weak Duality Theorem) Much less easy fact: (Strong Duality Theorem).

PRIMAL-DUAL INTERIOR METHODS FOR NONCONVEX NONLINEAR PROGRAMMING ANDERS FORSGRENyAND PHILIP E. GILLz SIAM J. OPTIM. °c Society for Industrial and Applied Mathematics Vol. 8, No. 4, pp.

{, November Abstract. primal and dual problem 1. primal & dual problems,their inter relationship prepared by: –khambhayata mayur – khant vijay – lad yash a.

submitted to: prof. Lecture 3: SVM dual, kernels and regression C19 Machine Learning Hilary A. Zisserman • Primal and dual forms • Linear separability revisted • Feature maps • Need to learn d parameters for primal, and N for dual • If NDual form only involves (xj>xk).