Paper review of ENGG*6140 Optimization Techniques Paper Review -- Interior-Point Methods for LP for LP Yanrong Hu Ce Yu Mar. 11, 2003

Outline General introduction The original algorithm description A variant of Karmarkar’s algorithm - a detailed simple example Another example More issues

Interior-point methods for linear programming:  The most dramatic new development in operations research (linear programming) during the 1980s.  Opened in 1984 by a young mathematician at AT&T Bell Labs, N. Karmarkar.  Polynomial-time algorithm and has great potential for solving huge linear programming problems beyond the reach of the simplex method.  Karmarkar’s method stimulated development in both interior-point and simplex methods. General introduction

Difference between Interior point methods and the simplex method The big difference between Interior point methods and the simplex method lies in the nature of trial solutions. General introduction Simplex method Interior-point method Trial solutions CPF (Corner Point Feasible) solutions Interior points (points inside the boundary of the feasible region) complexity worst case:# iterations can increase exponentially in the number of variables n: Polynomial time

Karmarkar’s original Algorithm Karmarkar assumes that the LP is given in Canonical form of the problem: Assumptions: To apply the algorithm to LP problem in standard form, a transformation is needed. Min Z = CX s.t. AX  b X  0

Karmarkar’s original Algorithm An example of transformation Min z = y1+y2 (z=cy) s.t. y1+2y2  2 (AY  b) y1, y2  0 Min z=5x1 + 5x2 s.t. 3x1+8x2+3x3-2x4=0 (AX=0) x1+ x2+ x3+ x4=1 (1X =1) x j  0, j=1,2,3,4 from standard form to canonical form

Karmarkar’s original Algorithm The principal idea: The algorithm creates a sequence of points having decreasing values of the objective function. In the step, the point is brought into the center of the simplex by a projective transformation.

The steps of Karmarkar’s algorithm are: Step 0. start with the solution point Step k. Define: and compute: and compute the step length parameters Karmarkar’s original Algorithm X is brought to the center by:

A Variant The Affine Variant of Karmarkar’s Algorithm: interior Concept 1: Shoot through the interior of the feasible region toward an optimal solution. Concept 2: Move in a direction that improves the objective function value fastest at the fastest feasible rate. Transform Concept 3: Transform the feasible region to place the current trail solution center near its center, thereby enabling a large improvement when concept 2 is implemented.

An Example Max Z= x1+ 2x2 s.t. x1+ x2  8 x1,x2  0 Optimal solution is (x1,x2)=(0,8) with Z=16 The Problem:

An Example  The direction is perpendiculars to (and toward) the objective function line. (3,4)=(2,2)+(1,2), where the gradient vector (1,2) is the gradient of the Objective Function. coefficient The gradient is the coefficient of the objective function. interior Concept 1: Shoot through the interior of the feasible region toward an optimal solution.  The algorithm begins with an initial solution that lies in interior the interior of the feasible region.  Arbitrarily choose (x1,x2)=(2,2) to be the initial solution. fastest Concept 2: Move in a direction that improves the objective function value at the fastest feasible rate. Max Z= x1+ 2x2 s.t. x1+ x2  8 x1,x2  0

An Example The augmented form can be written in general as: Max Z = c T X s.t. AX = b X  0 The algorithm actually operates on the augmented form. Max Z = x1+ 2x2 s.t. x1+ x2+ x3 = 8 x1,x2,x3  0 letting x3 be the slack Max Z = x1+ 2x2 s.t. x1+ x2  8 x1,x2  0 Initial solution: (2,2) Gradient of obj. fn.: (1,2) (2,2,4) (1,2,0)

An Example - Feasible region : the triangle - Optimum (0,8,0), Initial solution: (2,2,4) - Gradient of Obj. fn. : c T =[ 1, 2, 0 ] Show The augmented form graphically Projected Gradient Using Projected Gradient to implement Concept 1 & 2:  Adding the gradient to the initial leads to (3,4,4)= (2, 2, 4)+ (1,2,0) (infeasible)  To remain feasible, the algorithm project the point (3,4,4) down onto the feasible triangle.  To do so, the projected gradient – gradient projected onto the feasible region – is used.  Then the next trial solution move in the direction of projected gradient

An Interior-point Algorithm  a formula is available for computing the projected gradient: projection matrix : P = I-A T (AA T ) -1 A projected gradient: c p = Pc Projected Gradient Using Projected Gradient to implement Concept 1 & 2  now we are ready to move from (2,2,4) in the direction of c p to a new point:  determines how far we move. large   too close to the boundary small   more iterations we have chosen  =0.5, so the new trial solution move to: x = ( 2, 3, 3)

An Interior-point Algorithm Concept 3: Transform center Transform the feasible region to place the current trail solution near its center, thereby enabling a large improvement when concept 2 is implemented. Centering scheme for implementing Concept 3 Why -- The centering scheme keeps turning the direction of the projected gradient to point more nearly toward an optimal solution as the algorithm converges toward this solution. changing the scale How -- Simply changing the scale (units) for each of the variable so that the trail solution becomes equidistant from the constraint boundaries in the new coordinate system. Define D=diag{x}, bring x to the center in the new coordinate

An Interior-point Algorithm Initial trial solution (x1,x2,x3) =(2,2,4) In this new coordinate system, the problem becomes:

Summary and Illustration of the Algorithm Iteration 1. Given the initial solution (x1,x2,x3) = (2,2,4) Step1. given the current trial solution (x 1,x 2,…,x n ), set Summary of the General Procedure X is brought to the center (1,1,1) by:

Summary and Illustration of the Algorithm Summary of the General Procedure (cont.) Iteration 1. (cont.) To compute projected gradient:

Summary and Illustration of the Algorithm Iteration 1. (cont.) Summary of the General Procedure (cont.) Compute projected gradient:

Summary and Illustration of the Algorithm step4. define v as the absolute value of the negative component of c p having the largest value, so that v=|-2|=2. In this coordinate, the algorithm moves from the current trial solution to the next one. step4. Determine how far to move by identify v. Then make move by calculating Iteration 1. (cont.)

Summary and Illustration of the Algorithm this completes the iteration 1. The new solution will be used to start the next iteration. Step 5. Back to the original coordinate by calculating Step 5. In the original coordinate, the solution is Iteration 1. (cont.) Summary of the General Procedure (cont.)

Summary and Illustration of the Algorithm Iteration 2. Given the current trial solution X is brought to the center (1,1,1,) in the new coordinate, and move to (0.83, 1.4, 0.5), corresponds (2.08, 4.92,1.0) in the original coordinate

Summary and Illustration of the Algorithm Iteration 3. Given the current trial solution x=(2.08, 4.92, 1.0) X is brought to the center (1,1,1,) in the new coordinate, and move to ( 0.54,1.30,0.50), corresponds (1. 13, 6.37, 0.5) in the original coordinate

An Example Effect of rescaling of each iteration: Sliding the optimal solution toward (1,1,1) while the other BF solutions tend to slide away. AB CD

Summary and Illustration of the Algorithm More iterations. Starting from the current trial solution x, following the steps, x is moving toward the optimum (0,8). When the trial solution is virtually unchanged from the proceeding one, then the algorithm has virtually converged to an optimal solution. So stop.

Another Example The problem MAX Z = 5x1 + 4x2 ST6x1 + 4x2 =0 Augmented form MAX Z = 5x1 + 4x2 ST6x1 + 4x2 + x3=24 x1 + 2x2 + x4=6 -x1 + x2 + x5=1 x2 + x6=2 xj>=0, j=1,2,3,4,5,6

Another Example Starting from an initial solution x=(1,1,14,3,1,1) The trial solution at each iteration

More issues  Interior-point methods is designed for dealing with big problems. Although the claim that it’s much faster than the simplex method is controversy, many tests on huge LP problems show its outperformance.  After Karmarkar’s paper, many related methods have been developed to extend the applicability of Karmarkar’s algorithm, e.g. Infeasible interior points method -- remove the assumption that there always exits a nonempty interior. Methods applying to LP problems in standard form.

More issues Methods dealing with finding initial solution, and estimating the optimal solution. Methods working with primal-dual problems. Studies about moving step-long/short steps. Studies about efficient implementation and complexity of various methods.  Karmarkar’s paper not only started the development of interior point methods, but also encouraged rapid improvement of simplex methods.

Reference [1] N. Karmarkar, 1984, A New Polynomial - Time Algorithm for Linear Programming, Combinatorica 4 (4), 1984, p [2] M.J. Todd, (1994), Theory and Practice for Interior-point method, ORSA Jounal on Computing 6 (1), 1994, p [3] I. Lustig, R. Marsten, D. Shanno, (1993), Interior-point Methods for Linear Programming: Computational State of the Art, ORSA Journal on Computing, 6 (1), 1994, p [4] Hillier,Lieberman,Introduction to Operations Research (7th edition) [5] Taha, Operations Research: An Introduction (6th edition) [6] E.R. Barnes, 1986, A Variation on Karmarkar’s Algorithm for Sloving Linear Programming problems, Mathematical Programming 36, 1986, p [7] R.J. Vanderbei, M.S. Meketon and B.A. Freeman, A Modification of karmarkar’s Linear Programming Algorithm, Algorithmica: An International Journal in Computer Science 1 (4), 1986 p. 395 – 407. [8] D. Gay (1985) A Variant of Karmarkar’s Linear Programming Algorithm for Problems in Standard form, Mathematical Programming 37 (1987) more…

Yanrong Hu Ce Yu Mar Paper Review of ENGG*6140 Optimization Techniques