Convex Optimization — Boyd & Vandenberghe 5 Duality

What is primal in linear programming In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem

Relations between Primal and Dual

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

Primal Heuristics in SCIP

Primal Heuristics #define HEUR_TIMING SCIP_HEURTIMING_BEFORENODE 10 / 34 Heuristic Timings Primal Heuristics Domain Propagation Solve LP Pricing Separation

Convex Optimization — Boyd & Vandenberghe 5 Duality

primal problem minimize cTx subject to Ax b dual function g(λ)=inf x (c+ATλ)Tx−bTλ = ˆ −bTλ ATλ+c =0 −∞ otherwise dual problem maximize −bTλ subject to ATλ+c =0, λ 0 • from Slater’s condition: p⋆ =d⋆ if Ax˜≺ b for some x˜ • in fact, p⋆ =d⋆ except when primal and dual are infeasible Duality 5–12

4 Duality Theory

The primal-dual pair of LPs PDare related via the Weak Duality Theorem Theorem 4 1 (Weak Duality Theorem) If x 2 Rn is feasible for P and y 2 Rm is feasible for D, then cTx yTAx bTy Thus, if P is unbounded, then D is necessarily infeasible, and if D is unbounded, then P is necessarily infeasible

Unit 2: Section C A1 Sample answer Explain the part of Totem

religions Primal religions date back to 50,000 BCE The term religion has evolved overtime In our ancestor’s time it referred to the way in which ancient societies attached meaning to key moments in their lives There are 3 similar characteristics that define a primal religion All of

Institutional Meat Purchase Specifications

Fat Limitations : 8 7 1 : Beef Side Skeletal Diagram 24 : 3 4 1 Carcasses & Quarters 8 7 2 Beef Side Primal Diagram 25 3 4 2 Cuts, Roasts, Diced & Special Trim

[PDF] Convex Optimization — Boyd & Vandenberghe 5 Duality

What is primal in linear programming

Relations between Primal and Dual

Primal Heuristics in SCIP

Convex Optimization — Boyd & Vandenberghe 5 Duality

4 Duality Theory

Unit 2: Section C A1 Sample answer Explain the part of Totem

Institutional Meat Purchase Specifications