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FTEC 2101/ESTR2520 Optimization Methods Spring 2022
Online Midterm Examination.Quaranteed Success.Attained A+> Time allowed: 1.75 hours Time: 9:30-11:15am, March 11, 2022 Policy:
- Full marks will be given for completely solved problem with well justification. You will receive partial mark if you demonstrate sufficient understanding of the question and its solution, otherwise you will receive zero mark.
- The examination is open book and open notes, however, it must be done on your own. You should write your original solution, without soliciting any help from a third party. To ask for clarification on the midterm questions, please ‘raise your hand’ in Zoom. Problem 1. (25%) Consider the following linear programming problem in standard form: max − x 1 + 2 x 2 + 3 x 3 x 1 ,x 2 ,x 3 s.t. − x 1 + x 2 ≤ 3 − x 2 + 2 x 3 ≤ 2 x 1 − 3 x 3 ≤ 6 , x 1 ,x 2 ,x 3 ≥ 0. Apply the simplex method to find an optimal solution to the LP problem or to show that the LP problem is unbounded. You may initialize the simplex method with a solution satisfying ( x 1 ,x 2 ,x 3 ) = (0 , 0 , 0). Solution. We begin by introducing the slack variables and rewriting the problem in the canonical form: max z
x 1 ,x 2 ,x 3 ,s 1 ,s 2 ,s 3 ,z s.t. z + x 1 − 2 x 2 − 3 x 3 = 0 − x 1 + x 2 + s 1 = 3 − x 2 + 2 x 3 + s 2 = 2 x 1 − 3 x 3 + s 3 = 6 , x 1 ,x 2 ,x 3 ,s 1 ,s 2 ,s 3 ≥ 0. It is clear that we can take s 1 ,s 2 ,s 3 as the initial basis which will lead to a feasible CPF solution. We thus form the simplex table: BV z x 1 x 2 x 3 s 1 s 2 s 3 RHS z 1 1 -2 -3 0 0 0 0 s 1 0 -1 1 0 1 0 0 3 s 2 0 0 -1 2 0 1 0 2 s 3 0 1 0 -3 0 0 1 6 1 We select x 3 as the entering variable and s 2 as the leaving variable. Observe the following row opera- tions: − 0 2 −
3 R^2 − / 2 →→ R^2 00
2 R 3 +3 R 2 → R 3
Resulting in the table:
(^0 1 0 0 3) 1 0 1 / 2 0 1 0 0 3 / 2 1 9 B V z x 1 x 2 x 3 s 1 s 2 s 3 R HS z 1 1 -7/2 0 0 3/2 0 3 s 1
x 3
Solution: (a) Notice that the constraint max{ x 1 ,x 2 } ≤ 1 is equivalent to x 1 ≤ 1, and x 2 ≤ 1 simultaneously. The feasible region is thus sketched as: Applying the graphical method shows that the only optimal solution is (1 , 1) with an optimal objective value of 2. (b) The feasible region is sketched as: ✗✗ 2 I - ¥EEz >×' Applying the graphical method shows that the only optimal solution is (2 , 0) with an optimal objective value of 2. (c) Notice that satisfying the constraints min 0 is equivalent to having max{ x 1 ,x 2 } ≤ 1 ,x 1 ≥ 0 ,x 2 ≥ 0 OR. In other words, the feasible region is given by the union of the feasible sets in part (a) and (b). This can be sketched as: ✗✗ 2 ✗ ✗ 2 " % £ 1 "^ "
i*ITIÉm¥
' "' Applying the graphical method shows that there are two optimal solutions: (1 , 1) AND (2 , 0). Both have an optimal objective value of 2. Problem 3. (25%) Answer the following question on convex functions. (a) (5%) Let a ∈ R and b ≥ 0 be given. Consider the following set X = { x,y ∈ R : y > 0} Consider the function f : X → R , Evaluate the gradient vector ∇ f ( x,y ) and the Hessian matrix ∇^2 f ( x,y ). (b) (10%) Show that f ( x,y ) in part (a) is a convex function for all ( x,y ) ∈ X. (c) (10%) Set b > 0 and consider the function g : X → R , Show that g ( x,y ) is not a convex function. (Hint: check the non-PSD-ness of the Hessian matrix by fixing ( x,y ).) Solution. (a) The gradient and Hessian are given as: . (b) Our idea is to check the PSD-ness of ∇^2 f ( x,y ). Observing that which can be simplified as
- You spend less than or equal to 30 hours per week on studying.
- The grade point for the course FCEC2101 has to be greater than or equal to 3.0. Formulate an optimization problem to achieve the above goal. (b) (10%) Rewrite the optimization problem formulated in (a) as a linear program in standard form. (c) (10%) We consider a different case where the grade point for the courses is calculated according to a quantized system. In particular, suppose you spend xA hours per week on course A, then the grade point of course A is given by Q( xA ) such that 4 xA if 1 > ≥ 0 , then Q( xA ) = 0 difficulty of course A 4 xA if 2 > ≥ 1 , then Q( xA ) = 1 difficulty of course A 4 xA if 3 > ≥ 2 , then Q( xA ) = 2 difficulty of course A 4 xA if 4 > ≥ 3 , then Q( xA ) = 3 difficulty of course A 4 xA if ≥ 4 , then Q( xA ) = 4 difficulty of course A Formulate a mixed-integer optimization problem to design your study plan such that
- The objective is to maximize the GPA.
- You spend less than or equal to 30 hours per week on studying. Hint: You may first formulate the problem while keeping the Q(·) notation; also, observe that Q( xA ) = 0 or 1 or ··· or 4. You are not required to solve any of the optimization problems formulated in the above. Solution.
(a) The decision variables are: xA – amt. of time on EGGG3100 , xB – amt. of time on FCEC1200 , xC – amt. of time on FCEC The constraints are: . and The objective is to maximize w.r.t. xA,xB,xC for Thus, the overall optimization problem is given as: s.t. xA + xB + xC ≤ 30 , xA ≥ 0 , xB ≥ 0 , xC ≥ 0 , xC ≥ 15 / 4. (b) To rewrite the above optimization problem, we introduce the slack variables zA,zB,zC and observe the following equivalent form s.t. Replacing zA ≤ min{16 , 4 xA } by zA ≤ 16 , zA ≤ 4 xA , ..., lead to s.t. This is a standard form LP as desired. (c) We set the following as decision variables: xA – amt. of time on EGGG3100 , xB – amt. of time on FCEC1200 , xC – amt. of time on FCEC zA – GP of EGGG3100 , zB – GP of FCEC1200 , zC – GP of FCEC
