CPS 616 ITERATIVE IMPROVEMENTS 10-1

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1 CPS 66 ITERATIVE IMPROVEMENTS 0 - APPROACH Algorithm design technique for solving optimization problems Start with a feasible solution Repeat the following step until no improvement can be found: change the current feasible solution to a feasible solution with a better value of the objective function Return the last feasible solution as optimal Note: Tpicall, a change in a current solution is small (local search) Major difficult: Local optimum vs. global optimum EXAMPLE: SIMPLEX METHOD Linear Programming (LP) Problem: optimize a linear function of several variables subject to linear constraints: maimize (or minimize) c c n n subject to a i a in n (or or =) b i, i =,...,m 0,..., n 0 The function z = c c n n is called the objective function; constraints 0,..., n 0 are called nonnegativit constraints Possible Outcomes. Problem has a finite optimal solution, which ma not be unique. Problem could be unbounded: the objective function of maimization (minimization) LP problem is unbounded from above (below) on its feasible region. Problem could be infeasible: there are no points satisfing all the constraints, i.e. the constraints are contradictor Etreme Point Theorem: An LP problem with a nonempt bounded feasible region has an optimal solution; moreover, an optimal solution can alwas be found at an etreme point of the problem's feasible region

2 CPS 66 ITERATIVE IMPROVEMENTS 0 - Eample - Final Optimal Solution maimize + 5 subject to , 0 The Feasible region is the set of points defined b the constraints An Optimal Solution to the LP Problem is a point for which the value of the objective function is maimized.

3 CPS 66 ITERATIVE IMPROVEMENTS 0 - Eample - Unbounded problem Eample - Unfeasible problem maimize + 5 subject to , 0

4 CPS 66 ITERATIVE IMPROVEMENTS 0-4 Simple Method The classic method for solving LP maimization problems; one of the most important algorithms ever invented Invented b George Dantzig in 947 Based on the iterative improvement idea: Generates a sequence of adjacent points of the problem s feasible region with improving values of the objective function until no further improvement is possible Step 0: Initialization Step 0.: convert inequalities maimize + 5 maimize u + 0v subject to + 4 subject to + + u = v = 6 0, 0 0, 0, u 0, v 0 Variables u and v, transforming inequalit constraints into equalit constrains, are called slack variables Step 0.: calculate basic feasible solution A basic solution to a sstem of m linear equations in n unknowns (n m) is obtained b setting n m variables to 0 and solving the resulting sstem to get the values of the other m variables. The variables set to 0 are called nonbasic; the variables obtained b solving the sstem are called basic. A basic solution is called feasible if all its (basic) variables are nonnegative. Eample + + u = v = 6 (0, 0, 4, 6) is basic feasible solution (, are non basic; u, v are basic) There is a - correspondence between etreme points of LP s feasible region and its basic feasible solutions. Calculate value of function at that solution: u + 0v = 0

5 CPS 66 ITERATIVE IMPROVEMENTS 0-5 Simple Tableau representation of step 0. u v Basic u 0 4 variables v 0 6 Objective row Reverse coefficients value of z at (0,0,4,6) Simple Algorithm Step 0 [Initialization] Present a given LP problem in standard form and set up initial tableau. Step [Optimalit test] If all entries in the objective row are nonnegative stop: the tableau represents an optimal solution. Step [Find entering variable] Select (the most) negative entr in the objective row. Mark its column to indicate the entering variable and the pivot column. Step [Find departing variable] For each positive entr in the pivot column, calculate the θ-ratio b dividing that row's entr in the rightmost column b its entr in the pivot column. (If there are no positive entries in the pivot column stop: the problem is unbounded.) Find the row with the smallest θ-ratio, mark this row to indicate the departing variable and the pivot row. Step 4 [Form the net tableau] Divide all the entries in the pivot row b its entr in the pivot column. Subtract from each of the other rows, including the objective row, the new pivot row multiplied b the entr in the pivot column of the row in question. Replace the label of the pivot row b the variable's name of the pivot column and go back to Step.

6 CPS 66 ITERATIVE IMPROVEMENTS 0-6 Eample Simple Tableau Basic feasible solution z u v u 0 4 v 0 6 (0, 0, 4, 6) u v u 0 0 (0,,, 0) u v 0 0 (,, 0, 0) Notes on the Simple Method Finding an initial basic feasible solution ma pose a problem Theoretical possibilit of ccling Tpical number of iterations is between m and m, where m is the number of equalit constraints in the standard form Worse-case efficienc is eponential

7 CPS 66 ITERATIVE IMPROVEMENTS 0-7 Eample # Maimise + = For = =0 =0 +=8 +=0 +=0 = = -+= +=8 -+= +=8 =0 += +=0 =0 +=4 +=0 =0 =0 = = +=6 -+= +=5 +=8 -+= +=8 =0 +=0 =0 +=0 =0 =0 4 5

8 CPS 66 ITERATIVE IMPROVEMENTS 0-8 a b c d θ ratio a a = - -+ b b = +- +=8 c c = d d = 0-- MAX-- = MAX=+ a b c d θ ratio a a = b = +- +=8 c / c = d / d = 0-- MAX-- = MAX=+ a b c d θ ratio a = undef b = +- +=8 c c = d / d = 0-- MAX-- = MAX=+ a b c d 0 0 θ ratio a = b = +- +=8 b / c = d d = 0-- MAX-- = MAX=+ a b c d θ ratio / / 4 - a = / - - b = +- +=8 b c = a / / -/ d = 0-- MAX-- = MAX=+