Post on 02-Feb-2021
Hassan Hijazi AUSSOIS 2012
INVEX FORMULATIONS IN INTEGER PROGRAMMING
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09/01/2012
Convex functions 1
q Convex opHmizaHon: " Any staHonary point is opHmal
Invex functions 2
-1 -0,75 -0,5 -0,25 0 0,25 0,5 0,75 1 1,25
-0,25
0,25
0,5
0,75
1
Definition (Hanson 1981) 4
Simple characterization 5
Constrained Optimization 6
We need to look at the Lagrangian funcHon:
¨ Modeling disjuncHve constraints featuring unbounded variables
¨ Invex formulaHons for a facility locaHon problem
Invex formulations in Integer Programs 7
Unbounded disjunction 8
¨ How to formulate the constraint:
¨ x must remain unbounded!
Ø Now, we only need to model:
Unbounded disjunction 9
z
y
The big-M formulation 10
y
z
Convex hull formulation 11
Lifting 12
z
y
ϒ
A second order cone constraint 13
Cplex 14
CPLEX 12.2.0.0: best soluGon found, primal-‐dual infeasible; objecGve 3.749997256 50 barrier iteraGons No basis. x = 3.75
A hidden hypothesis: constraint qualification
15
If a point x* saHsfies a constraint qualificaHon condiHon, it is opHmal if and only if it saHsfies the KKT condiHons. q LICQ: the gradients of the acHve inequality constraints and the gradients of the equality constraints are linearly independent at x*. q Slater condiGons: there exists a point x’ such that gi(x’) < 0 for all gi acHve in x*.
A hidden hypothesis: constraint qualification
16
Invex formulation 17
z
y
ϒ
Invex formulation 18
It works! 19
Using IPOPT open source solver, Interior point method
With Ipopt 20
Total CPU secs in IPOPT (w/o funcGon evaluaGons) = 0.003 Total CPU secs in NLP funcGon evaluaGons = 0.000 EXIT: OpGmal SoluGon Found. Ipopt 3.8.3: OpGmal SoluGon Found x = 4 gamma = 0 y = 0 z = 0
Concentrator placement in Smart Energy Grids
21
Concentrator
Smart meter
Mathematical modeling 22
Mathematical modeling 23
Mathematical modeling 24
Mathematical modeling 25
Mathematical modeling 26
Mathematical modeling 27
Example 28
minimize cost: 100*z1 + 100*z2 + 25*x11 + 35*x12 + 50*x21 + 35*x22; subject to demand1: 1 -‐ x11 -‐ x12
LP relaxation 29
CPLEX 12.2.0.0: opGmal soluGon; objecGve 172.5 4 dual simplex iteraGons (0 in phase I) z1 = 0.5 z2 = 0.5 x11 = 0.5 x12 = 0.5 x21 = 0.5 x22 = 0.5
Example 30
minimize cost: 100*z1^2 + 100*z2^2 + 25*y11 + 35*y12 + 50*y21 + 35*y22; subject to demand1: 1 -‐ x11 -‐ x12
Invex relaxation 31
ObjecGve...............: 168.859 Ipopt 3.8.3: OpGmal SoluGon Found z1 = 0.5 z2 = 0.559344 x11 = 1 x12 = 4.55569e-‐09 x21 = 5.02689e-‐09 x22 = 1
Finding a feasible solution
|I| |J| Bonmin’s best Invex
rdata1 15 250 2300 39
rdata2 20 250 >3000 43
rdata3 30 250 >3000 120
rdata4 40 250 >3000 150
rdata5 100 250 >3000 300
32
Bonmin 1.5 using CBC-‐IPOPT, Hme limit = 3000 sec
33