A New Method for Quick Solving Quadratic Assignment Problem CO 769. Paper presentation. by Zhongzhen Zhang

Focus. “A New Method for Quickly Solving Quadratic Assignment Problems” Key concept. “ Pivoting Algorithms ” Linear Programming well established in Quadratic Programming Follows from. “ An efficient method for solving the local minimum of indefinite quadratic programming” More work: same person & same topic. Book (in Chinese,2004) Convex Programming: Pivoting Algorithms for Portfolio Selection and Network Optimization 0. Before intro CO 769An algorithm for solving Quadratic Assignment Problems

1. Intro: Pivoting CO 769An algorithm for solving Quadratic Assignment Problems Start with a set of vectors Definition. The subset is called a basis if it is linearly independent and maximal i.e. Assume is a basis. Represent this situation with the table Basic vectors Nonbasic vectors

1. Pivoting CO 769An algorithm for solving Quadratic Assignment Problems non zero swapped How to do it? What comes out? Pivoting. Swap one basic vector with a nonbasic vector When it can be done?

1. Pivoting CO 769An algorithm for solving Quadratic Assignment Problems non zero Pivoting. Swap one basic vector with a nonbasic vector Update the table ! … … … …

2.Solving systems of linear equalities&inequalities CO 769An algorithm for solving Quadratic Assignment Problems Solve: B can be empty l can be 0 Take large M>0: Set up the “pivoting table”: System defined by n+m vectors:

2.Solving systems of linear equalities&inequalities CO 769An algorithm for solving Quadratic Assignment Problems = ≥ ≥ Let x1 be the solution of: measure the offset x1 is a solution

3. Preprocessing: move the equalities in the basis. CO 769An algorithm for solving Quadratic Assignment Problems = Assume is <0 1.Find a positive element no such thing: infeasible problem 2.Do a pivoting on a positive element for example a11 Theorem. >0 ≥≥ = = ≥ ≥ ≥ = = ≥ ≥ ≥≥ = ≥

3. Preprocessing: move the equalities in the basis. CO 769An algorithm for solving Quadratic Assignment Problems = Assume is >0 1.Find a negative element no such thing: infeasible problem 2.Do a pivoting on a negative element for example ≥≥ = = ≥ ≥ ≥ = ≥ ≥ ≥ ≥ ≥== Theorem. Once an equality goes in the basis it stays there ! ! !

3. Preprocessing: move the equalities in the basis. CO 769An algorithm for solving Quadratic Assignment Problems Assume is =0 1.If everything =0 the equality is redundant. Just throw it away. 2.Else do a pivoting on a nonzero element = ≥ ≥ ≥ ≥ ≥==

3. Main iterations CO 769An algorithm for solving Quadratic Assignment Problems ≥ = ≥ ≥ ≥ ≥ =≥≥ assume <0 1.Find a positive element no such thing: infeasible 2.Do a pivoting on it Pivoting rules: the smallest deviation rule, the largest distance rule, the smallest index rule etc. Theorem. When cycling doesn’t occur.

4. Quadratic programming CO 769An algorithm for solving Quadratic Assignment Problems What I’m about to do can be done for general quadratic problems For simplicity, I’ll work directly with the quadratic problem that will come from QAP Same ideas

4. Quadratic programming CO 769An algorithm for solving Quadratic Assignment Problems Consider:where As before: Set up KKT for this:

4. Quadratic programming CO 769An algorithm for solving Quadratic Assignment Problems Consider:where KKT Step 1: get rid of γ’s 0≤0≤

4. Quadratic programming. Preprocessing. CO 769An algorithm for solving Quadratic Assignment Problems New KKT The unknowns Ignore this! ≥≥≥≥ ≥ ≥ = = Step 1. Preprocessing. ! ! Must become basic ! !

4. QP. Preprocessing. CO 769An algorithm for solving Quadratic Assignment Problems New KKT The unknowns Ignore this! ≥≥≥≥ ≥ ≥ = = Step 1. Preprocessing. ! ! Must become basic ! !

4. QP. Preprocessing. CO 769An algorithm for solving Quadratic Assignment Problems KKT The unknowns Ignore this! ≥=≥≥ ≥ ≥ ≥ = Step 1. Preprocessing. ! ! Must become basic ! ! Choose pivot! !Don’t destroy other complementarity!

4. QP. Preprocessing. CO 769An algorithm for solving Quadratic Assignment Problems KKT The unknowns Ignore this! ≥=≥≥ ≥ ≥ ≥ = Step 1. Preprocessing. ! ! Must become basic ! !

4. QP. Preprocessing. CO 769An algorithm for solving Quadratic Assignment Problems =0 or 1 Preprocessing leads to a feasible solution Choosing pivots in preprocessing can be done in more than one way. Pivot selection plays a role in the implementation. If a choice fails, another is pursued. =0 or 1 or -1 Remark 1. The method described in the paper is heuristic. 2. Attempts to gain insight are made. 3. No insights for pivot selection in preprocessing.

4. QP. Main Iterations: Local search. CO 769An algorithm for solving Quadratic Assignment Problems Local search: What happens: The offset column must be updated. Feasibility can be lost!! Not any local search is good. Goal: Decrease in the objective! Nothing else. No other insight.

4. QP. Main Iterations: Pivoting. CO 769An algorithm for solving Quadratic Assignment Problems In general for QP: pivoting is done such that complementarity is preserved ! Pivoting can destroy the feasibility ! Keep feasibility all the time Pivoting modifies the objective ! Heuristic goal: pivot to decrease the objective Forward pivoting. Assume the pivot is produces a feasible solution Feasibility Objective Decrease

4. QP. Main Iterations: Pivoting. CO 769An algorithm for solving Quadratic Assignment Problems In general for QP: pivoting is done such that complementarity is preserved ! Pivoting can destroy the feasibility ! Keep feasibility all the time Pivoting modifies the objective ! Heuristic goal: pivot to decrease the objective Backward pivoting. Assume the pivot is produces a feasible solution Feasibility Objective Decrease

4. QP. Main Iterations: Pivoting. CO 769An algorithm for solving Quadratic Assignment Problems In general for QP: pivoting is done such that complementarity is preserved ! Pivoting can destroy the feasibility ! Keep feasibility all the time Pivoting modifies the objective ! Heuristic goal: pivot to decrease the objective Par pivoting. Assume the pivot is Feasibility No Decrease in Objective

4. QP. Main Iterations: Pivoting. CO 769An algorithm for solving Quadratic Assignment Problems Par pivoting. Assume the pivot is Feasibility No Decrease in Objective This pivoting: against heuristic. Experiments show: equivalent with +

4. QP. Main Iterations: Pivoting. CO 769An algorithm for solving Quadratic Assignment Problems “The contraction effect of pivoting operations occurs to us the formation of substance that is under certain conditions or control while the formation of a good solution of QAP is depending on the appropriate control of pivoting operations. It is hopeful to discover the deep mechanism of QAP and develop more efficient computing method.” Thanks! Z.Zhang