Outline. Linear Programming (LP): Simplex Search. Simplex: An Extreme-Point Search Algorithm. Basic Solutions

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1 Outline Linear Programming (LP): Simplex Search Benoît Chachuat McMaster University Department of Chemical Engineering ChE 4G03: Optimization in Chemical Engineering 1 Basic Solutions 2 The Simplex Algorithm 3 Two-Phase Simplex 4 Weird Events For additional details, see Rardin (1998), Chapter Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 1 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 2 / 35 Simplex: An Extreme-Point Search Algorithm Basic Solutions Starting point Outstanding Questions: No further improvement 1 How to characterize extreme feasible points? Consider only adjacent extreme points for improvement direction Move along the edge that yields the greatest rate of improvement Move until another extreme point has been reached 2 How to move to an adjacent extreme feasible point? 3 How to start at an extreme feasible point? Check if further improvement is possible: if yes, continue; else, stop LP standard form: min z = c T x x s.t. Ax = b x 0 x j = jth decision variable c j = objective function coefficient of x j a i,j = constraint coefficient of x j in the ith main constraint b i = right-hand side (RHS) constant term of main constraint i m = number of main constraints n = number of decision variables Definition: A basic solution to a linear program in standard form is one obtained by fixing just enough variables to = 0 that the model s equality constraints can be solved uniquely for the remaining variable values Those variables fixed at zero are called nonbasic variables The ones obtained by solving the equalities are called basic variables Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 3 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 4 / 35

2 Computing Basic Solutions Class Exercise: Consider the following linear program: min s.t. 4 + = , 0 Question: Compute the basic solution corresponding to and basic. Existence of Basic Solutions Can basic solutions be formed by setting any collection of nonbasic variables to zero? Not so... Important Property A basic solution exists if and only if the system of equality constraints corresponding to basic variables has a unique solution Math Refresher: 1 How do you calculate the determinant of an m-by-m matrix A? 2 Give a necessary and sufficient condition for the system of m linear equations Ax = b to have a unique solution in IR m. Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 5 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 6 / 35 Checking Existence of Basic Solutions Class Exercise: The following are the constraints of a standard form LP: 4 8 x 3 = 15 2 = 10,,x 3 0 Question: Do basic solutions exist for the sets of basic variables: (a) B = {, }: (b) B = { }: (c) B = {,x 3 }: Basic Feasible Solutions and Extreme Points Definition A basic feasible solution to an LP in standard form is a basic solution that satisfies all nonnegativity constraints Class Exercise: Is the solution corresponding to and x 3 basic in the previous exercise a basic feasible one? Fundamental Result The basic feasible solutions of a linear program in standard form are exactly the extreme points of its feasible region Since algebraic tests exist to check basic feasible solutions, we now are in a position to characterize extreme points (i.e. potential optimal solutions) algebraically This is quite a significant result! Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 7 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 8 / 35

3 Identifying Basic Feasible Solutions Class Exercise: The following are the constraints of an LP model: Identifying Basic Feasible Solutions (cont d) , 0 Questions: 1 Graph the feasible region and indicate the extreme points 2 Rewrite the LP in standard form by introducing slack variables x 3, x 4 and x 5 in the main constraints 1 to 3, respectively. 3 Compute the basic solutions corresponding to the following sets of basic variables, and determine which are basic feasible solutions: (a) B 1 = {x 3,x 4,x 5 }; (b) B 2 = {,,x 4 }; (c) B 3 = {,,x 5 }. Verify that each basic feasible solution corresponds to an extreme point in the graph. How about basic infeasible solutions? Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 9 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 10 / 35 Simplex Algorithm: Principles The simplex algorithm is a variant of improving search, elegantly adapted to the peculiarities of LP in standard form Every step of the simplex visits an extreme point of the LP feasible domain Getting Started! An improving search begins by choosing a starting feasible solution, and simplex requires an extreme point: Initial Basic Solution Simplex search begins at an extreme point of the feasible region (i.e., at a basic feasible solution to the model in standard form) Standard Display: max + s.t , 0 = Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 11 / 35 Example: Try with basic variables B (0) = {,x 4,x 5 }? 3 (x 5 0) + 0 (x 3 0) Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 12 / 35 x1 0 x (0) 0 x1 2 (x4 0)

4 Simplex Directions We want simplex to follow edge directions joining current extreme points to adjacent ones: Each edge direction follows a line determined by all but one of the active constraints at the current extreme point But we know that the active constraints at a basic feasible solution correspond to the nonnegativity constraints on nonbasic variables Class Exercise: Starting from x (0), how to follow the edge defined by: 1 the active constraint + = 0? 2 the active constraint = 0? 3 (x 5 0) + 0 (x 3 0) Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 13 / 35 x1 0 x (0) 0 x1 2 (x4 0) Simplex Directions (cont d) There is one simplex direction for each nonbasic variable! 1 Let B be the current basic feasible solution, and consider the nonbasic variable x j / B: x j = 1, and x k = 0, x k / B,k j 2 We want to remain feasible as we move along an edge. What is the change in basic variables x k B needed to preserve the equality constraints Ax = b? Ax = b A(x + x) = b Constructing Simplex Direction } = A x = 0 Simplex directions are constructed by increasing a single nonbasic variable, leaving other nonbasic unchanged, and computing the (unique) corresponding change in basic variables necessary to preserve equality constraints Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 14 / 35 Constructing Simplex Directions Class Exercise: Calculate the simplex directions corresponding to the basic feasible solution B (0) = {,x 4,x 5 } for the following standard-form LP: Constructing Simplex Directions (cont d) x for x for x 3 Simplex dir. for x 3 x1 0 x (0) 3 (x 5 0) + 0 (x 3 0) 0 x1 2 (x4 0) Simplex dir. for Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 15 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 16 / 35

5 Which Simplex Direction to Follow? Checking Improvement of Simplex Directions Our next task is to see whether any of the simplex directions improve the objective function f (x) = c T x Moving along a simplex direction x for nonbasic x j incurs a change in objective equal to c j is the so-called reduced cost Improving Simplex Directions c j = c T x The simplex direction x increasing nonbasic x j is improving if: c j > 0, for a maximize problem c j < 0, for a minimize problem Class Exercise: Determine which of the simplex directions computed previously are improving for the specified maximizing objective function: x for c 1 = x for x 3 c 3 = Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 17 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 18 / 35 How Long to Follow the Simplex Direction? For the next move, simplex can adopt any simplex direction x that improves the objective function The next issue is How far? What step size λ in the direction x? The limit on step size λ can only come from violating a nonnegativity constraint. Why? Maximum Acceptable Step If any component is negative in improving simplex direction x at current basic solution x (k), simplex search uses the maximum acceptable step, x (k) j λ = min : x j < 0 x j Making the Next Move Class Exercise: Determine the maximum step and new solution in the selected improving simplex direction Our new solution is: c = 2 λ = x (1) x (0) + λ x = Why only negative components? Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 19 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 20 / 35

6 Making the Next Move (cont d) 3 (x 5 0) λ = 2 x (1) = x (0) + λ x x1 0 x (0) + 0 (x 3 0) x 0 x1 2 (x4 0) Updating the Basic Solution To continue the algorithm, we need to find a new basic feasible solution Active nonnegativity constraints tell us how: Update After each move of simplex search, the nonbasic variable generating the chosen simplex becomes basic any one of the (possibly several) basic variables fixing the step size becomes nonbasic Class Exercise: Determine the new basic feasible solution in the previous exercise B (1) Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 21 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 22 / 35 Rudimentary Simplex Search for LP Step 0: Initialization Choose any starting feasible basic solution B (0), construct the starting point x (0), and let index k 0. Step 1: Simplex Directions Construct the simplex directions x associated with increasing each nonbasic x j, and compute the corresponding reduced cost c j = c T x If no simplex direction is improving, stop current solution x (k) is globally optimal Otherwise, choose any improving simplex direction x, and denote the entering basic variable x e Rudimentary Simplex Search for LP Step 2: Step Size If there is no limit on feasible moves in simplex direction x, stop The model is unbounded. Otherwise, choose the step size λ so that λ = min and denote the leaving variable x l Step 3: Update { } (k) x j : x j < 0 x j Compute the new solution point and basic solution: x (k+1) x (k) + λ x B (k+1) B (k) {x e } \ {x l } Increment index k k + 1 and return to Step 1 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 23 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 24 / 35

7 Determining the Optimum Class Exercise: Continue the simplex search from the new basic feasible solution B (1) = {,,x 5 }: B (1) B B N N B x (1) c T x (1) = x for.... Generating a Starting Basic Feasible Solution In most problems, we have to search for a starting basic feasible solution, before Simplex search can be applied. LP standard form: min z = c T x x s.t. Ax = b x 0 m = number of main constraints n = number of decision variables Basic Feasible Solution: (n m) elements of x are set to zero nonbasics remaining m elements are nonnegative and satisfy Ax = b basics Idea Formulate an artificial LP model, the solution of which provides a basic feasible solution to the original LP Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 25 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 26 / 35 Artificial LP Model Formulation and Solution Artificial LP An artificial LP is constructed as one that minimizes the sum of constraint violation for the equality constraints in the original LP model Step 1: Introduce new nonnegative artificial variables x n+1,...,x n+m such that: a 1,1 + +a 1,n x n ±x n+1 = b 1 a 2,1 + +a 2,n x n ±x n+2 = b =. a m,1 + +a m,n x n ±x n+m = b m Add the artificial variable x n+j if b j 0 Subtract the artificial variable x n+j otherwise Artificial LP Model Formulation and Solution Step 2: Formulate the following artificial LP with (n + m) variables: min v = x n x n+m x s.t. a 1,1 + +a 1,n x n ±x n+1 = b =. a m,1 + +a m,n x n ±x n+m = b m,...,x n,x n+1,...,x n+m 0 The artificial LP is in standard form Why? B (0) = {x n+1,...,x n+m } is a basic feasible solution Why? Step 3: Solve the auxiliary LP, starting from B (0) = {x n+1,...,x n+m } Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 27 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 28 / 35

8 Artificial LP Model Formulation and Solution Class Exercise: Formulate an artificial LP model to identify a starting basic feasible solution to the following LP model in standard form: max + s.t. + x 3 = 0 + x 4 = 2 + x 5 = 3,,x 3,x 4,x 5 0 Two-Phase Simplex Search 1 Phase I: Apply simplex search to the artificial LP model 2 Infeasibility: If Phase I search terminates with a minimum having artificial sum v > 0, stop The original LP model is infeasible Otherwise, use the final Phase I basic solution to identify a starting feasible basic solution for the original LP model 3 Phase II: Apply simplex search, starting from the identified basic feasible solution, to compute an optimal solution to the original standard-form LP model or prove that it is unbounded Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 29 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 30 / 35 Degeneracy It is not always the case that a better extreme point is encountered at each iteration! With simplex search, progress at some iterations is typically interspersed with periods of no advance: Degeneracy solution value (max) iterations A basic feasible solution to a standard-form LP is degenerate if nonnegativity constraints for some basic variables are active (i.e., more constraints are active than strictly needed to define an extreme point) Degeneracy (cont d) Multiple Choices of Basic Variables In the presence of degeneracy, several sets of basic variables compute the same basic solution Graphical Illustration: All 5 inequalities B, C, H, I and G are active at x (1) the corresponding slack variables have value = 0 Any 3 of these 5 inequalities define extreme point x (1) Hence, 5 3 = 2 basic variables have value = 0 The solution is degenerate! H B x (1) I A x 3 G C J D K E L F Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 31 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 32 / 35

9 Investigating the Effects of Degeneracy Class Exercise: Consider the following LP model, with given starting basic feasible solution B (0) 1 Explain why the solution is degenerate 2 Show that x for is an improving simplex direction 3 Calculate the step length along that direction 4 Update the basic/nonbasic variables in B (1) and calculate x (1) B (0) N N B B B x for c 1 = λ = B (1) x (1) c T x (1) = Zero-Length Simplex Steps Simplex directions that decrease basic variables already = 0 in a degenerate solution may produce moves with steps λ = 0 Rules When degenerate solutions cause the simplex algorithm to compute a step λ = 0, 1 the basic/nonbasic variables should be changed accordingly, 2 computations should be continued as if a positive step had been taken Simplex computations will normally escape a sequence of degenerate moves (changing basic representations eventually produces a direction along which positive progress can be achieved) Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 33 / 35 Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 34 / 35 The Final Words on Simplex Search Convergence with Simplex If each iteration yields a positive step λ > 0, simplex search will stop after finitely many iterations, with either an optimal solution or an indication of unboundedness maximum number of extreme points = This is a finite number, yet it can be very large! Degeneracy and Cycling n! (n m)!m! Cycling can occur with degenerate solutions: a sequence of degenerate moves may return to a basic/nonbasic configuration it has already visited! It is usually safe to assume that cycling will not occur in applied LP models, and thus that simplex search will converge finitely Benoît Chachuat (McMaster University) LP: Simplex Search 4G03 35 / 35

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