5.3 Mixed Integer Nonlinear Programming Models. A Typical MINLP Model.
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Transcript of 5.3 Mixed Integer Nonlinear Programming Models. A Typical MINLP Model.
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5.3 Mixed Integer Nonlinear Programming Models
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A Typical MINLP Model
min
. .
,where, is the vector of continuous variables; is the vector of integer (usually binary) variables; is a matrix; and are sets
Tz f
s t
X Y
X Y
x c y
h x 0
g x My 0x y
xy M
.
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RemarksThe y’s are typically chosen to control the continuo
us variables x by either forcing one (or more) variable to be zero or by allowing them to assume positive values.
The choice of y should be done in such a way that y appears linearly, because then the problem is much easier to solve.
The set X is specified by bounds and other inequalities involving x only, whereas Y is defined by conditions that the components of y be binary or integer, plus other inequalities or equations involving y only.
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Branch-and-Bound Method• The same BB method used to solve MILP can b
e used to solve MINLP.• The only difference is that for MINLP problems t
he relaxed subproblems at the nodes of the BB tree are continuous variable NLPs and must be solved by NLP methods.
• BB methods are guaranteed to solve linear or nonlinear problems if allowed to continue until the gap between upper and lower bounds reaches zero, provided that a global optimum is found for each relaxed subproblem at each node of the BB tree.
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Sufficient Conditions of Convexity of Each Relaxed Subproblem
1. The objective function f(x) is convex.2. Each component of h(x) is linear.3. Each component of g(x) is convex over th
e set X.4. The set X is convex.5. The set Y is determined by linear constrai
nts and the integer restrictions on y.
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Example: Optimal Selection of Processes
This problem involves the manufacture of a chemical C in process 1 that uses raw material B. B can either be purchased or produced via processes 2 or 3, both of which use chemical A as a raw material.
We want to determine which processes to use and their production levels in order to maximize profit.
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Constraints of Example Problem
1 0.9 1, 2 ln 1 2 , 3 1.2 ln 1 3
1 2 3
2, 3, 1, 2, 3, , 1 0
1, 2, 3 0,1
C B B A B A
B B B BP
A A B B B BP C
Y Y Y
1. Conversion
2. Mass balance for B
3. Nonnegativity conditions for continuous variables
4. Integer constraints
5. Maximum d1 1
2 4 2, 3 5 3, 1 2 1
C
B Y B Y C Y
emand for C
6. Limits on plant capacity
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Objective Function of Example Problem
• Income from product sales: 13C1• Expense for the purchase of B: 7BP• Expense for the purchase of A: 1.8(A2+A3)• Annualized investment for the 3 processes:
(3.5Y1+2C1)+(Y2+B2)+(1.5Y3+1.2B3)• The objective function is profit (PR) to be
maximized: PR=11C1-3.5Y1-Y2-1.5Y3-B2-1.2B3-7BP-1.8A2-1.8A3
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Solving MINLP Using Outer Approximation (OA)
Each major iteration of OA involves solving 2 subproblems:
1. a continuous variable nonlinear program (NLP), and
2. A mixed-integer linear program (MILP).
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NLP Subproblem
The NLP subproblem at major iteration , NLP( ),is formed by fixing the integer variables at some set
of values, say Y, and optimizing over the continuousvariables:
Problem NLP( ):
min. .
k
k
k
T k
k
fs t
y
y
yc y x
h x =
Note that the optimal objective value is the UPPER BOUNDon the MINLP optimal value.
k
X
0
g x + My 0x
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MILP Subproblem
At iteration , it is fromed by linearizing all nonlinear functions
about the optimal solutions of each of the subproblem NLP ,
1, 2, , , and keeping all of these linearized constraints.
Let be the
i
i
k
i k
y
x
solution of NLP . The MILP subproblem at iteration is
min. .
, 1, 2, ,
, 1, 2, ,
, 1, 2, ,
,
i
T
i T i
i T i
i T i i
k
zs t
z f f i k
i k
i k
X Y
y
c x
x x - x
h x h x - x 0
g x g x - x My 0
x y
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The Role of New Variable in the MILP Sub-problem
min. .
is equivalent to minimizing .
T
T
zs tz f
f
c y
x
c y x
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OA Algorithm• Duran and Grossman (1986) showed that if the c
onvexity assumptions hold, then the optimal value of MILP subproblem is an LOWER BOUND on the optimal MINLP objective value.
• Because a new set of linear constraints is added at each iteration, this lower bound increases (or remains the same) at each iteration.
• Under the convexity assumptions, the upper and lower bounds converge to the true optimal MINLP value in a finite number of iterations.