Mixed-integer Convex Representability

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Juliusz Wasilewski
4 lat temu
Przeglądów:

1 Mixed-integer Convex Representability Juan Pablo Vielma Massachuse=s Ins?tute of Technology Joint work with Miles Lubin and Ilias Zadik INFORMS Annual Mee?ng, Phoenix, AZ, November, 2018.

2 Mixed-Integer Convex Optimization (MICONV) min f(x) s.t. x 2 C x i 2 Z i 2 I convex f and C.

3 What Can MICONV Model? Two sheet hyperbola? Spherical shell? x 2 R 2 :1+x 2 1 apple x 2 2 x 2 R 2 :1applekxk apple2 Integer points in parabola { ", " $ " Z}? The set of ) ) matrices with rank 0? Set of prime numbers?

4 MICONV Can Model Any Finite Union of (Closed) Convex Sets % Let! ",,! % be closed convex set. A MICONV formulation of & )*" ; ; & ), + ) cone! ) {1} 5 {1,, 6} 7 +) 9 ). 5 {1,, 6} % & ) 7 )*" % )*" & ) = &, + ) = 1, = {0,1} %, 9 %, & ) R A 5 {1,, 6}! ) :

5 What Can MICONV Model? Two sheet hyperbola? Spherical shell? ü Simple MICONV Formula3on x 2 R 2 :1+x 2 1 apple x 2 2 x 2 R 2 :1applekxk apple2 Integer points in parabola { ", " $ " Z}? The set of ) ) matrices with rank 0? Set of prime numbers?

6 A Simple Obstruction for MICONV Formulations S cannot have a MICONV formula<on if there exists: There exist infinite u + v 2 Spherical shell R S /2 S 8u, v 2 R, u6= v x 2 R 2 :1applekxk apple2 R s.t.

7 What Can MICONV Model? Two sheet hyperbola? ü Spherical shell? x 2 R 2 :1+x 2 1 apple x 2 2 x 2 R 2 :1applekxk apple2 Integer points in parabola { ", " $ " Z}? The set of ) ) matrices with rank 0? Set of prime numbers? Does have non-convex polynomial MIP formula8on

8 <latexit sha1_base64="+3waeaprsewoksxf2jrtlptakxc=">aaacmhicbvbnswmxem36bf2qevqslepfrztf0kpoqy8ktopnxbjpwopz7dbjiu3sn+tfn6ixbuw8+ivmtj1o9cha470zzuyfsedauo6rmzi6nj4xotwdm5mdm1/ily5vdjqoyso0epg6dihmgktwntwidhkrrsjasivg9ijzl+6y0jys56yds1pimpi3ocxgsn7+gouwmmnx3kfbxmfj2tu8971rr7uobwtbti12ohdclyxhbzczzqu3elkhnxxmjcqhmtdbkf51/xzblbk9wl8edugbdhdq559wpajjykshgmhd MICONV = Structured Countably Infinite Unions of Convex Sets N q (x 1 2z) 2 + x 2 2 z 1, z 2 Z apple 1 1/z, Can be strange unions, e.g. : Infinite number of shapes P 1 P 2 P 3 P 4 Can be REALLY strange: Dense discrete set np 2x jp 2x k o : x 2 N [0, 1] Unbounded Integer Variables k(z 1,z 1 )k 2 apple z 2 +1, k(z 2,z 2 )k 2 apple 2z 1, x 1 = y 1 z 2, k(z 1,z 1 )k 2 apple y 1, k(y 1,y 1 )k 2 apple 2z 1, z 2 Z 2

9 Ques%ons?

10 A Simple Lemma for non-micp Representability Obstruc<on for MICP representability of S : u + v infinite R S s.t. /2 S 8u, v 2 R, u6= v 2 Proof: Assume for contradic<on there exists M such that: S =proj x M \ R n+p Z d (u, y u, z u ) 2 M (v, y v, z v ) 2 M ) z u + z v 2 /2 Z d z u z v (mod 2) component-wise ) z u + z v 2 2 Z d component-wise parity classes = 2 d < R = 1 ) (

11 <latexit sha1_base64="+3waeaprsewoksxf2jrtlptakxc=">aaacmhicbvbnswmxem36bf2qevqslepfrztf0kpoqy8ktopnxbjpwopz7dbjiu3sn+tfn6ixbuw8+ivmtj1o9cha470zzuyfsedauo6rmzi6nj4xotwdm5mdm1/ily5vdjqoyso0epg6dihmgktwntwidhkrrsjasivg9ijzl+6y0jys56yds1pimpi3ocxgsn7+gouwmmnx3kfbxmfj2tu8971rr7uobwtbti12ohdclyxhbzczzqu3elkhnxxmjcqhmtdbkf51/xzblbk9wl8edugbdhdq559wpajjykshgmhd <latexit sha1_base64="/vxooivfx/micwegget0ukiabbc=">aaac03iclvjnb9qwehxcr9vlowsce7fyixgavzjus5tdpype4ic0lwxbarnwjunsrtpoze+ajvgcxju/x43fwj/am+4bthxgjetpb96bgy+dvojr8lwfjnvt+o2bg5tbnvu379zd7t67f6llwle2oauo1vlknbncsglweoysu MICONV = Structured Countably Infinite Unions of Convex Sets Can be strange unions, e.g. : Infinite number of shapes P 1 P 2 P 3 P 4 N q (x 1 2z) 2 + x 2 2 z 1, z 2 Z apple 1 1/z, There exist an increasing funcdon h such that: P z R 2 regular h(z)-gon S = [ 1 has MICONV formuladon z=1 P z Equal volume Finite # of Shapes

12 <latexit sha1_base64="qfbajf0uvyy62repzyuphtv5vsy=">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 Sets with MICONV Formulations can be REALLY Strange Dense discrete set np jp k o 2x 2x : x 2 N [0, 1] k(z 1,z 1 )k 2 apple z 2 +1, k(z 2,z 2 )k 2 apple 2z 1, x 1 = y 1 z 2, k(z 1,z 1 )k 2 apple y 1, k(y 1,y 1 )k 2 apple 2z 1, z 2 Z 2 Non-Periodic Set of naturals n x 2 N : p 2x jp 2x k /2 ", 1 p 2" o k(x 1,x 1 )k 2 apple x 2 + ", k(x 2,x 2 )k 2 apple 2x 1 +2", x 2 Z 2 + "God made the integers, all else is the work of man - Leopold Kronecker

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