Mixed Integer Nonlinear Programming - UZH - ZCCFE · A Popular MINLP Method Dantzig’s Two-Phase...

Mixed Integer Nonlinear Programming (MINLP)

Sven Leyffer

MCS DivisionArgonne National Lableyffer@mcs.anl.gov

Jeff Linderoth

ISE DepartmentLehigh Universityjtl3@lehigh.edu

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Overview

1 Introduction, Applications, and Formulations

2 Classical Solution Methods

3 Modern Developments in MINLP

4 Implementation and Software

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Part I

Introduction, Applications, and Formulations

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The Problem of the Day

Mixed Integer Nonlinear Program (MINLP)minimize

x ,yf (x , y)

subject to c(x , y) ≤ 0x ∈ X , y ∈ Y integer

f , c smooth (convex) functions

X ,Y polyhedral sets, e.g. Y = y ∈ [0, 1]p | Ay ≤ by ∈ Y integer ⇒ hard problem

f , c not convex ⇒ very hard problem

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x ,yf (x , y)

X ,Y polyhedral sets, e.g. Y = y ∈ [0, 1]p | Ay ≤ b

y ∈ Y integer ⇒ hard problem

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x ,yf (x , y)

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x ,yf (x , y)

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Why the MI?

We can use 0-1 (binary) variables for a variety of purposes

Modeling yes/no decisionsEnforcing disjunctionsEnforcing logical conditionsModeling fixed costsModeling piecewise linear functions

If the variable is associated with a physical entity that is indivisible,then it must be integer

Number of aircraft carriers to to produce. Gomory’s Initial Motivation

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Why the MI?

We can use 0-1 (binary) variables for a variety of purposes

Modeling yes/no decisionsEnforcing disjunctionsEnforcing logical conditionsModeling fixed costsModeling piecewise linear functions

If the variable is associated with a physical entity that is indivisible,then it must be integer

Number of aircraft carriers to to produce. Gomory’s Initial Motivation

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A Popular MINLP Method

Dantzig’s Two-Phase Method for MINLP Adapted by Leyffer and Linderoth

1 Convince the user that he or she does not wish to solve a mixedinteger nonlinear programming problem at all!

2 Otherwise, solve the continuous relaxation (NLP) and round off theminimizer to the nearest integer.

For 0− 1 problems, or those in which the |y | is “small”, thecontinuous approximation to the discrete decision is not accurateenough for practical purposes.

Conclusion: MINLP methods must be studied!

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Example: Core Reload Operation (Quist, A.J., 2000)

max. reactor efficiency after reloadsubject to diffusion PDE & safety

diffusion PDE ' nonlinear equation⇒ integer & nonlinear model

avoid reactor becoming sub-critical

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max. reactor efficiency after reloadsubject to diffusion PDE & safety

diffusion PDE ' nonlinear equation⇒ integer & nonlinear model

avoid reactor becoming overheated

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look for cycles for moving bundles:e.g. 4 → 6 → 8 → 10i.e. bundle moved from 4 to 6 ...

model with binary xilm ∈ 0, 1xilm = 1⇔ node i has bundle l of cycle m

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AMPL Model of Core Reload Operation

Exactly one bundle per node:

L∑l=1

M∑m=1

xilm = 1 ∀i ∈ I

AMPL model:var x I,L,M binary ;

Bundle i in I: suml in L, m in M x[i,l,m] = 1 ;

Multiple Choice: One of the most common uses of IP

Full AMPL model c-reload.mod atwww.mcs.anl.gov/~leyffer/MacMINLP/

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Gas Transmission Problem (De Wolf and Smeers, 2000)

Belgium has no gas!

All natural gas is importedfrom Norway, Holland, orAlgeria.

Supply gas to all demandpoints in a network in aminimum cost fashion.

Gas is pumped through thenetwork with a series ofcompressors

There are constraints on thepressure of the gas withinthe pipe

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Belgium has no gas!

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Belgium has no gas!

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Pressure Loss is Nonlinear

Assume horizontal pipes andsteady state flows

Pressure loss p across a pipe isrelated to the flow rate f as

p2in − p2

out =1

Ψsign(f )f 2

Ψ: “Friction Factor”

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Gas Transmission: Problem Input

Network (N,A). A = Ap ∪ Aa

Aa: active arcs have compressor. Flow rate can increase on arcAp: passive arcs simply conserve flow rate

Ns ⊆ N: set of supply nodes

ci , i ∈ Ns : Purchase cost of gas

s i , s i : Lower and upper bounds on gas “supply” at node i

pi, pi : Lower and upper bounds on gas pressure at node i

si , i ∈ N: supply at node i .

si > 0⇒ gas added to the network at node isi < 0⇒ gas removed from the network at node i to meet demand

fij , (i , j) ∈ A: flow along arc (i , j)

f (i , j) > 0⇒ gas flows i → jf (i , j) < 0⇒ gas flows j → i

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Gas Transmission: Problem Input

Network (N,A). A = Ap ∪ Aa

Aa: active arcs have compressor. Flow rate can increase on arcAp: passive arcs simply conserve flow rate

Ns ⊆ N: set of supply nodes

ci , i ∈ Ns : Purchase cost of gas

s i , s i : Lower and upper bounds on gas “supply” at node i

pi, pi : Lower and upper bounds on gas pressure at node i

si , i ∈ N: supply at node i .

si > 0⇒ gas added to the network at node isi < 0⇒ gas removed from the network at node i to meet demand

fij , (i , j) ∈ A: flow along arc (i , j)

f (i , j) > 0⇒ gas flows i → jf (i , j) < 0⇒ gas flows j → i

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Gas Transmission Model

min∑j∈Ns

subject to ∑j |(i ,j)∈A

fij = si ∀i ∈ N

sign(fij)f2ij −Ψij(p

2i − p2

j ) = 0 ∀(i , j) ∈ Ap

sign(fij)f2ij −Ψij(p

2i − p2

j ) ≥ 0 ∀(i , j) ∈ Aa

si ∈ [s i , s i ] ∀i ∈ Npi ∈ [p

i, pi ] ∀i ∈ N

fij ≥ 0 ∀(i , j) ∈ Aa

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Your First Modeling Trick

Don’t include nonlinearities or nonconvexities unless necessary!

Replace p2i ← ρi

sign(fij)f2ij −Ψij(ρi − ρj) = 0 ∀(i , j) ∈ Ap

f 2ij −Ψij(ρi − ρj) ≥ 0 ∀(i , j) ∈ Aa

ρi ∈ [√pi,√

pi ] ∀i ∈ N

This trick only works because1 p2

i terms appear only in the bound constraints2 Also fij ≥ 0 ∀(i , j) ∈ Aa

This model is nonconvex: sign(fij)f2ij is a nonconvex function

Some solvers do not like sign

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Replace p2i ← ρi

f 2ij −Ψij(ρi − ρj) ≥ 0 ∀(i , j) ∈ Aa

ρi ∈ [√pi,√pi ] ∀i ∈ N

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Replace p2i ← ρi

f 2ij −Ψij(ρi − ρj) ≥ 0 ∀(i , j) ∈ Aa

ρi ∈ [√pi,√pi ] ∀i ∈ N

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Dealing with sign(·): The NLP Way

Use auxiliary binary variables to indicate direction of flow

Let |fij | ≤ F ∀(i , j) ∈ Ap

1 fij ≥ 0 fij ≥ −F (1− zij)0 fij ≤ 0 fij ≤ Fzij

Note thatsign(fij) = 2zij − 1

Write constraint as

(2zij − 1)f 2ij −Ψij(ρi − ρj) = 0.

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Special Ordered Sets

Sven thinks this ’NLP trick’ is pretty cool

It is not how it is done in De Wolf and Smeers (2000).

Heuristic for finding a good starting solution, then a localoptimization approach based on a piecewise-linear simplex method

Another (similar) approach involves approximating the nonlinearfunction by piecewise linear segments, but searching for the globallyoptimal solution: Special Ordered Sets of Type 2

If the “multidimensional” nonlinearity cannot be removed, resort toSpecial Ordered Sets of Type 3

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Portfolio Management

N: Universe of asset to purchase

xi : Amount of asset i to hold

B: Budget

minx∈R|N|+

u(x) |

∑i∈N

xi = B

Markowitz: u(x)def= −αT x + λxTQx

α: Expected returnsQ: Variance-covariance matrix of expected returnsλ: Risk aversion parameter

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Portfolio Management

N: Universe of asset to purchase

xi : Amount of asset i to hold

B: Budget

minx∈R|N|+

u(x) |

∑i∈N

xi = B

Markowitz: u(x)def= −αT x + λxTQx

α: Expected returnsQ: Variance-covariance matrix of expected returnsλ: Risk aversion parameter

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More Realistic Models

b ∈ R|N| of “benchmark” holdings

Benchmark Tracking: u(x)def= (x − b)TQ(x − b)

Constraint on E[Return]: αT x ≥ r

Limit Names: |i ∈ N : xi > 0| ≤ K

Use binary indicator variables to model the implicationxi > 0⇒ yi = 1Implication modeled with variable upper bounds:

xi ≤ Byi ∀i ∈ N∑i∈N yi ≤ K

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More Realistic Models

b ∈ R|N| of “benchmark” holdings

Benchmark Tracking: u(x)def= (x − b)TQ(x − b)

Constraint on E[Return]: αT x ≥ r

Limit Names: |i ∈ N : xi > 0| ≤ K

Use binary indicator variables to model the implicationxi > 0⇒ yi = 1Implication modeled with variable upper bounds:

xi ≤ Byi ∀i ∈ N∑i∈N yi ≤ K

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Even More Models

Min Holdings: (xi = 0) ∨ (xi ≥ m)

Model implication: xi > 0⇒ xi ≥ mxi > 0⇒ yi = 1⇒ xi ≥ mxi ≤ Byi , xi ≥ myi ∀i ∈ N

Round Lots: xi ∈ kLi , k = 1, 2, . . .xi − ziLi = 0, zi ∈ Z+ ∀i ∈ N

Vector h of initial holdings

Transactions: ti = |xi − hi |Turnover:

∑i∈N ti ≤ ∆

Transaction Costs:∑

i∈N ci ti in objective

Market Impact:∑

i∈N γi t2i in objective

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Even More Models

∑i∈N ti ≤ ∆

Market Impact:∑

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Even More Models

∑i∈N ti ≤ ∆

Market Impact:∑

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Multiproduct Batch Plants (Kocis and Gross-

mann, 1988)

M: Batch Processing Stages

N: Different Products

H: Horizon Time

Qi : Required quantity of product i

tij : Processing time product i stage j

Sij : “Size Factor” product i stage j

Bi : Batch size of product i ∈ N

Vj : Stage j size: Vj ≥ SijBi ∀i , jNj : Number of machines at stage j

Ci : Longest stage time for product i : Ci ≥ tij/Nj ∀i , j

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Multiproduct Batch Plants (Kocis and Gross-

mann, 1988)

M: Batch Processing Stages

N: Different Products

H: Horizon Time

Qi : Required quantity of product i

tij : Processing time product i stage j

Sij : “Size Factor” product i stage j

Bi : Batch size of product i ∈ N

Vj : Stage j size: Vj ≥ SijBi ∀i , jNj : Number of machines at stage j

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Multiproduct Batch Plants

min∑j∈M

αjNjVβjj

Vj − SijBi ≥ 0 ∀i ∈ N,∀j ∈ MCiNj ≥ tij ∀i ∈ N,∀j ∈ M∑

BiCi ≤ H

Bound Constraints on Vj ,Ci ,Bi ,Nj

Nj ∈ Z ∀j ∈ M

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Modeling Trick #2

Horizon Time and Objective Function Nonconvex. :-(

Sometimes variable transformations work!

vj = ln(Vj), nj = ln(Nj), bi = ln(Bi ), ci = lnCi

min∑j∈M

αjeNj+βjVj

s.t. vj − ln(Sij)bi ≥ 0 ∀i ∈ N,∀j ∈ Mci + nj ≥ ln(τij) ∀i ∈ N,∀j ∈ M∑

i∈NQie

Ci−Bi ≤ H

(Transformed) Bound Constraints on Vj ,Ci ,Bi

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Modeling Trick #2

min∑j∈M

αjeNj+βjVj

s.t. vj − ln(Sij)bi ≥ 0 ∀i ∈ N, ∀j ∈ Mci + nj ≥ ln(τij) ∀i ∈ N,∀j ∈ M∑

i∈NQie

Ci−Bi ≤ H

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Modeling Trick #2

min∑j∈M

αjeNj+βjVj

s.t. vj − ln(Sij)bi ≥ 0 ∀i ∈ N, ∀j ∈ Mci + nj ≥ ln(τij) ∀i ∈ N,∀j ∈ M∑

i∈NQie

Ci−Bi ≤ H

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How to Handle the Integrality?

But what to do about the integrality?

1 ≤ Nj ≤ N j ∀j ∈ M,Nj ∈ Z ∀j ∈ M

nj ∈ 0, ln(2), ln(3), . . . ...

1 nj takes value ln(k)0 Otherwise

nj −K∑

ln(k)Ykj = 0 ∀j ∈ M

K∑k=1

Ykj = 1 ∀j ∈ M

This model is available at http://www-unix.mcs.anl.gov/

~leyffer/macminlp/problems/batch.mod

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A Small Smattering of Other Applications

Chemical Engineering Applications:

process synthesis (Kocis and Grossmann, 1988)batch plant design (Grossmann and Sargent, 1979)cyclic scheduling (Jain, V. and Grossmann, I.E., 1998)design of distillation columns (Viswanathan and Grossmann, 1993)pump configuration optimization (Westerlund, T., Pettersson, F. andGrossmann, I.E., 1994)

Forestry/Paper

production (Westerlund, T., Isaksson, J. and Harjunkoski, I., 1995)trimloss minimization (Harjunkoski, I., Westerlund, T., Porn, R. andSkrifvars, H., 1998)

Topology Optimization (Sigmund, 2001)

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Part II

Classical Solution Methods

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Classical Solution Methods for MINLP

1 Classical Branch-and-Bound

2 Outer Approximation & Benders Decomposition3 Hybrid Methods

LP/NLP Based Branch-and-BoundIntegrating SQP with Branch-and-Bound

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Branch-and-Bound

Solve relaxed NLP (0 ≤ y ≤ 1 continuous relaxation). . . solution value provides lower bound

Branch on yi non-integral

Solve NLPs & branch until1 Node infeasible ... •2 Node integer feasible ... ⇒ get upper bound (U)

3 Lower bound ≥ U ...⊗

y = 0i

dominated by upper bound

infeasible

integer feasibleetc.

Search until no unexplored nodes on tree

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Variable Selection for Branch-and-BoundAssume yi ∈ 0, 1 for simplicity ...(x , y) fractional solution to parent node; f = f (x , y)

1 maximal fractional branching: choose yi closest to 12

maximin(1− yi , yi )

2 strong branching: (approx) solve all NLP children:

f+/−i ←

minimize

x ,yf (x , y)

subject to c(x , y) ≤ 0x ∈ X , y ∈ Y , yi = 1/0

branching variable yi that changes objective the most:

min(f +

i , f−i )

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Variable Selection for Branch-and-BoundAssume yi ∈ 0, 1 for simplicity ...(x , y) fractional solution to parent node; f = f (x , y)

1 maximal fractional branching: choose yi closest to 12

maximin(1− yi , yi )

2 strong branching: (approx) solve all NLP children:

f+/−i ←

minimize

x ,yf (x , y)

subject to c(x , y) ≤ 0x ∈ X , y ∈ Y , yi = 1/0

branching variable yi that changes objective the most:

min(f +

i , f−i )

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Node Selection for Branch-and-Bound

Which node n on tree T should be solved next?1 depth-first search: select deepest node in tree

minimizes number of NLP nodes storedexploit warm-starts (MILP/MIQP only)

2 best estimate: choose node with best expected integer soln

minn∈T

fp(n) +∑

i :yi fractional

mine+i (1− yi ), e

−i yi

where fp(n) = value of parent node, e+/−i = pseudo-costs

summing pseudo-cost estimates for all integers in subtree

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Node Selection for Branch-and-Bound

Which node n on tree T should be solved next?1 depth-first search: select deepest node in tree

minimizes number of NLP nodes storedexploit warm-starts (MILP/MIQP only)

2 best estimate: choose node with best expected integer soln

minn∈T

fp(n) +∑

i :yi fractional

mine+i (1− yi ), e

−i yi

where fp(n) = value of parent node, e+/−i = pseudo-costs

summing pseudo-cost estimates for all integers in subtree

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Outer Approximation (Duran and Grossmann, 1986)

Motivation: avoid solving huge number of NLPs• Exploit MILP/NLP solvers: decompose integer/nonlinear part

Key idea: reformulate MINLP as MILP (implicit)• Solve alternating sequence of MILP & NLP

NLP subproblem yj fixed:

NLP(yj)

minimize

xf (x , yj)

subject to c(x , yj) ≤ 0x ∈ X

Main Assumption: f , c are convex

(y )jNLP

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• let (xj , yj) solve NLP(yj)• linearize f , c about (xj , yj) =: zj• new objective variable η ≥ f (x , y)• MINLP (P) ≡ MILP (M)

minimizez=(x ,y),η

subject to η ≥ fj +∇f Tj (z − zj) ∀yj ∈ Y

0 ≥ cj +∇cTj (z − zj) ∀yj ∈ Y

x ∈ X , y ∈ Y integer

SNAG: need all yj ∈ Y linearizations

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(Mk): lower bound (underestimate convex f , c)NLP(yj): upper bound U (fixed yj)

NLP( ) subproblemylinearizationNLP gives

MILP findsnew y

MILP infeasible?

MILP master program

⇒ stop, if lower bound ≥ upper bound

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Outer Approximation & Benders Decomposition

Take OA cuts for zj := (xj , yj) ... wlog X = Rn

η ≥ fj +∇f Tj (z − zj) & 0 ≥ cj +∇cTj (z − zj)

sum with (1, λj) ... λj multipliers of NLP(yj)

η ≥ fj + λTj cj + (∇fj +∇cjλj)T (z − zj)

KKT conditions of NLP(yj) ⇒ ∇x fj +∇xcjλj = 0... eliminate x components from valid inequality in y

⇒ η ≥ fj + (∇y fj +∇ycjλj)T (y − yj)

NB: µj = ∇y fj +∇ycjλj multiplier of y = yj in NLP(yj)References: (Geoffrion, 1972)

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LP/NLP Based Branch-and-Bound

AIM: avoid re-solving MILP master (M)

Consider MILP branch-and-bound

interrupt MILP, when yj found⇒ solve NLP(yj) get xj

linearize f , c about (xj , yj)⇒ add linearization to tree

continue MILP tree-search

... until lower bound ≥ upper bound

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integerfeasible

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need access to MILP solver ... call back exploit good MILP (branch-cut-price) solver (Akrotirianakis et al., 2001) use Gomory cuts in tree-search

preliminary results: order of magnitude faster than OA same number of NLPs, but only one MILP

similar ideas for Benders & Extended Cutting Plane methods

recent implementation by CMU/IBM group

References: (Quesada and Grossmann, 1992)

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Integrating SQP & Branch-and-Bound

AIM: Avoid solving NLP node to convergence.

Sequential Quadratic Programming (SQP)→ solve sequence (QPk) at every node

minimize

dfk +∇f Tk d + 1

2dTHkd

subject to ck +∇cTk d ≤ 0xk + dx ∈ X

yk + dy ∈ Y .

Early branching:After QP step choose non-integral yk+1

i , branch & continue SQPReferences: (Borchers and Mitchell, 1994; Leyffer, 2001)

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SNAG: (QPk) not lower bound⇒ no fathoming from upper bound

minimized

fk +∇f Tk d + 12d

yk + dy ∈ Y .

Remedy: Exploit OA underestimating property (Leyffer, 2001):

add objective cut fk +∇f Tk d ≤ U − ε to (QPk)

fathom node, if (QPk) inconsistent

NB: (QPk) inconsistent and trust-region active ⇒ do not fathom

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SNAG: (QPk) not lower bound⇒ no fathoming from upper bound

minimized

fk +∇f Tk d + 12d

yk + dy ∈ Y .

Remedy: Exploit OA underestimating property (Leyffer, 2001):

add objective cut fk +∇f Tk d ≤ U − ε to (QPk)

fathom node, if (QPk) inconsistent

NB: (QPk) inconsistent and trust-region active ⇒ do not fathom

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Comparison of Classical MINLP Techniques

Summary of numerical experience

1 Quadratic OA master: usually fewer iterationMIQP harder to solve

2 NLP branch-and-bound faster than OA... depends on MIP solver

3 LP/NLP-based-BB order of magnitude faster than OA. . . also faster than B&B

4 Integrated SQP-B&B up to 3× faster than B&B' number of QPs per node

5 ECP works well, if function/gradient evals expensive

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Part III

Modern Developments in MINLP

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Modern Methods for MINLP

1 Formulations

RelaxationsGood formulations: big M ′s and disaggregation

2 Cutting Planes

Cuts from relaxations and special structuresCuts from integrality

3 Handling Nonconvexity

EnvelopesMethods

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Relaxations

z(S)def= minx∈S f (x)

z(T )def= minx∈T f (x)

Independent of f , S ,T :z(T ) ≤ z(S)

If x∗T = arg minx∈T f (x)

And x∗T ∈ S , then

x∗T = arg minx∈S f (x)

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Relaxations

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Relaxations

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UFL: Uncapacitated Facility LocationFacilities: J

Customers: Imin

∑j∈J

fjxj +∑i∈I

∑j∈J

fijyij

∑j∈J

yij = 1 ∀i ∈ I∑i∈I

yij ≤ |I |xj ∀j ∈ J (1)

OR yij ≤ xj ∀i ∈ I , j ∈ J (2)

Which formulation is to be preferred?

I = J = 40. Costs random.

Formulation 1. 53,121 seconds, optimal solution.Formulation 2. 2 seconds, optimal solution.

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Customers: Imin

∑j∈J

fjxj +∑i∈I

∑j∈J

fijyij

∑j∈J

yij = 1 ∀i ∈ I∑i∈I

yij ≤ |I |xj ∀j ∈ J (1)

OR yij ≤ xj ∀i ∈ I , j ∈ J (2)

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Customers: Imin

∑j∈J

fjxj +∑i∈I

∑j∈J

fijyij

∑j∈J

yij = 1 ∀i ∈ I∑i∈I

yij ≤ |I |xj ∀j ∈ J (1)

OR yij ≤ xj ∀i ∈ I , j ∈ J (2)

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Customers: Imin

∑j∈J

fjxj +∑i∈I

∑j∈J

fijyij

∑j∈J

yij = 1 ∀i ∈ I∑i∈I

yij ≤ |I |xj ∀j ∈ J (1)

OR yij ≤ xj ∀i ∈ I , j ∈ J (2)

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Valid Inequalities

Sometimes we can get a better formulation by dynamicallyimproving it.

An inequality πT x ≤ π0 is a valid inequality for S ifπT x ≤ π0 ∀x ∈ S

Alternatively: maxx∈SπT x ≤ π0

Thm: (Hahn-Banach). Let S ⊂ Rn bea closed, convex set, and let x 6∈ S .Then there exists π ∈ Rn such that

πT x > maxx∈SπT x S

xπTx = π0

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Valid Inequalities

Sometimes we can get a better formulation by dynamicallyimproving it.

An inequality πT x ≤ π0 is a valid inequality for S ifπT x ≤ π0 ∀x ∈ S

Alternatively: maxx∈SπT x ≤ π0

Thm: (Hahn-Banach). Let S ⊂ Rn bea closed, convex set, and let x 6∈ S .Then there exists π ∈ Rn such that

πT x > maxx∈SπT x S

xπTx = π0

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Nonlinear Branch-and-Cut

Consider MINLPminimize

x ,yf Tx x + f Ty y

subject to c(x , y) ≤ 0y ∈ 0, 1p, 0 ≤ x ≤ U

Note the Linear objective

This is WLOG:

min f (x , y) ⇔ min η s.t. η ≥ f (x , y)

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It’s Actually Important!

We want to approximate the convex hull of integer solutions, butwithout a linear objective function, the solution to the relaxationmight occur in the interior.

No Separating Hyperplane! :-(

min(y1 − 1/2)2 + (y2 − 1/2)2

s.t. y1 ∈ 0, 1, y2 ∈ 0, 1

η ≥ (y1 − 1/2)2 + (y2 − 1/2)2

(y1, y2)

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It’s Actually Important!

We want to approximate the convex hull of integer solutions, butwithout a linear objective function, the solution to the relaxationmight occur in the interior.

No Separating Hyperplane! :-(

min(y1 − 1/2)2 + (y2 − 1/2)2

s.t. y1 ∈ 0, 1, y2 ∈ 0, 1

η ≥ (y1 − 1/2)2 + (y2 − 1/2)2

(y1, y2)

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Valid Inequalities From Relaxations

Idea: Inequalities valid for a relaxation are valid for original

Generating valid inequalities for a relaxation is often easier.

Separation Problem over T:Given x ,T find (π, π0) suchthat πT x > π0,πT x ≤ π0∀x ∈ T

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Simple Relaxations

Idea: Consider one row relaxations

If P = x ∈ 0, 1n | Ax ≤ b, then for any row i ,Pi = x ∈ 0, 1n | aTi x ≤ bi is a relaxation of P.

If the intersection of the relaxations is a good approximation to thetrue problem, then the inequalities will be quite useful.

Crowder et al. (1983) is the seminal paper that shows this to be truefor IP.

MINLP: Single (linear) row relaxations are also valid ⇒ sameinequalities can also be used

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Simple Relaxations

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Simple Relaxations

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Simple Relaxations

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Simple Relaxations

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Knapsack Covers

K = x ∈ 0, 1n | aT x ≤ b

A set C ⊆ N is a cover if∑

j∈C aj > b

A cover C is a minimal cover if C \ j is not a cover ∀j ∈ C

If C ⊆ N is a cover, then the cover inequality∑j∈C

xj ≤ |C | − 1

is a valid inequality for S

Sometimes (minimal) cover inequalities are facets of conv(K )

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Knapsack Covers

K = x ∈ 0, 1n | aT x ≤ b

A set C ⊆ N is a cover if∑

j∈C aj > b

A cover C is a minimal cover if C \ j is not a cover ∀j ∈ C

If C ⊆ N is a cover, then the cover inequality∑j∈C

xj ≤ |C | − 1

is a valid inequality for S

Sometimes (minimal) cover inequalities are facets of conv(K )

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Other Substructures

Single node flow: (Padberg et al., 1985)

x ∈ R|N|+ , y ∈ 0, 1|N| |∑j∈N

xj ≤ b, xj ≤ ujyj ∀ j ∈ N

Knapsack with single continuous variable: (Marchand and Wolsey,

x ∈ R+, y ∈ 0, 1|N| |∑j∈N

ajyj ≤ b + x

Set Packing: (Borndorfer and Weismantel, 2000)

S =y ∈ 0, 1|N| | Ay ≤ e

A ∈ 0, 1|M|×|N|, e = (1, 1, . . . , 1)T

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The Chvatal-Gomory Procedure

A general procedure for generating valid inequalities for integerprograms

Let the columns of A ∈ Rm×n be denoted by a1, a2, . . . anS = y ∈ Zn

+ | Ay ≤ b.1 Choose nonnegative multipliers u ∈ Rm

+2 uTAy ≤ uTb is a valid inequality (

∑j∈N uTajyj ≤ uTb).

j∈NbuTajcyj ≤ uTb (Since y ≥ 0).4∑

j∈NbuTajcyj ≤ buTbc is valid for S since buTajcyj is an integer

Simply Amazing: This simple procedure suffices to generate everyvalid inequality for an integer program

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Extension to MINLP (Cezik and Iyengar, 2005)

This simple idea also extends to mixed 0-1 conic programmingminimizezdef= (x ,y)

subject to Az K by ∈ 0, 1p, 0 ≤ x ≤ U

K: Homogeneous, self-dual, proper, convex cone

x K y ⇔ (x − y) ∈ K

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Gomory On Cones (Cezik and Iyengar, 2005)

LP: Kl = Rn+

SOCP: Kq = (x0, x) | x0 ≥ ‖x‖SDP: Ks = x = vec(X ) | X = XT ,X p.s.d

Dual Cone: K∗ def= u | uT z ≥ 0 ∀z ∈ K

Extension is clear from the following equivalence:

Az K b ⇔ uTAz ≥ uTb ∀u K∗ 0

Many classes of nonlinear inequalities can be represented as

Ax Kq b or Ax Ks b

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LP: Kl = Rn+

Ax Kq b or Ax Ks b

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LP: Kl = Rn+

Ax Kq b or Ax Ks b

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Using Gomory Cuts in MINLP (Akrotirianakis et al., 2001)

LP/NLP Based Branch-and-Bound solves MILP instances:minimizezdef= (x ,y),η

subject to η ≥ fj +∇f Tj (z − zj) ∀yj ∈ Y k

0 ≥ cj +∇cTj (z − zj) ∀yj ∈ Y k

Create Gomory mixed integer cuts from

η ≥ fj +∇f Tj (z − zj)

0 ≥ cj +∇cTj (z − zj)

Akrotirianakis et al. (2001) shows modest improvements

Research Question: Other cut classes?

Research Question: Exploit “outer approximation” property?

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0 ≥ cj +∇cTj (z − zj) ∀yj ∈ Y k

0 ≥ cj +∇cTj (z − zj)

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0 ≥ cj +∇cTj (z − zj) ∀yj ∈ Y k

0 ≥ cj +∇cTj (z − zj)

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Disjunctive Cuts for MINLP (Stubbs and Mehrotra, 1999)

Extension of Disjunctive Cuts for MILP: (Balas, 1979; Balas et al., 1993)

Continuous relaxation (zdef= (x , y))

Cdef= z |c(z) ≤ 0, 0 ≤ y ≤ 1, 0 ≤ x ≤ U

C def= conv(x ∈ C | y ∈ 0, 1p)

def= z ∈ C |yj = 0/1

letMj(C )def=

z = λ0u0 + λ1u1

λ0 + λ1 = 1, λ0, λ1 ≥ 0u0 ∈ C 0

j , u1 ∈ C 1j

⇒ Pj(C ) := projection of Mj(C ) onto z

continuousrelaxation

⇒ Pj(C ) = conv (C ∩ yj ∈ 0, 1) and P1...p(C ) = C

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Cdef= z |c(z) ≤ 0, 0 ≤ y ≤ 1, 0 ≤ x ≤ U

def= z ∈ C |yj = 0/1

letMj(C )def=

z = λ0u0 + λ1u1

λ0 + λ1 = 1, λ0, λ1 ≥ 0u0 ∈ C 0

j , u1 ∈ C 1j

convexhull

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Cdef= z |c(z) ≤ 0, 0 ≤ y ≤ 1, 0 ≤ x ≤ U

def= z ∈ C |yj = 0/1

letMj(C )def=

z = λ0u0 + λ1u1

λ0 + λ1 = 1, λ0, λ1 ≥ 0u0 ∈ C 0

j , u1 ∈ C 1j

integerfeasible

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Disjunctive Cuts: Example

minimizex ,y

x | (x − 1/2)2 + (y − 3/4)2 ≤ 1,−2 ≤ x ≤ 2, y ∈ 0, 1

C 0j C 1

z = (x , y)

xGiven z with yj 6∈ 0, 1 find separatinghyperplane

minimize

z‖z − z‖

subject to z ∈ Pj(C )

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Disjunctive Cuts ExampleC 0j C 1

z = (x , y)

z∗def= arg min ‖z − z‖

s.t. λ0u0 + λ1u1 = zλ0 + λ1 = 1(

−0.160

)≤ u0 ≤

)(−0.47

)≤ u1 ≤

)λ0, λ1 ≥ 0

NONCONVEX

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z = (x , y)

z∗def= arg min ‖z − z‖2

s.t. λ0u0 + λ1u1 = zλ0 + λ1 = 1(

−0.160

)≤ u0 ≤

)(−0.47

)≤ u1 ≤

)λ0, λ1 ≥ 0

NONCONVEX

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z = (x , y)

z∗def= arg min ‖z − z‖∞

s.t. λ0u0 + λ1u1 = zλ0 + λ1 = 1(

−0.160

)≤ u0 ≤

)(−0.47

)≤ u1 ≤

)λ0, λ1 ≥ 0

NONCONVEX

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z = (x , y)

z∗def= arg min ‖z − z‖

s.t. λ0u0 + λ1u1 = zλ0 + λ1 = 1(

−0.160

)≤ u0 ≤

)(−0.47

)≤ u1 ≤

)λ0, λ1 ≥ 0

NONCONVEX

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What to do? (Stubbs and Mehrotra, 1999)

Look at the perspective of c(z)

P(c(z), µ) = µc(z/µ)

Think of z = µz

Perspective gives a convex reformulation of Mj(C ): Mj(C ), where

(z , µ)

∣∣∣∣∣∣µci (z/µ) ≤ 00 ≤ µ ≤ 10 ≤ x ≤ µU, 0 ≤ y ≤ µ

c(0/0) = 0 ⇒ convex representation

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What to do? (Stubbs and Mehrotra, 1999)

Look at the perspective of c(z)

P(c(z), µ) = µc(z/µ)

Think of z = µz

Perspective gives a convex reformulation of Mj(C ): Mj(C ), where

(z , µ)

∣∣∣∣∣∣µci (z/µ) ≤ 00 ≤ µ ≤ 10 ≤ x ≤ µU, 0 ≤ y ≤ µ

c(0/0) = 0 ⇒ convex representation

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Disjunctive Cuts Example

∣∣∣∣∣∣∣∣µ[(x/µ− 1/2)2 + (y/µ− 3/4)2 − 1

]≤ 0

−2µ ≤ x ≤ 2µ0 ≤ y ≤ µ0 ≤ µ ≤ 1

C 0j C 1

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Example, cont.

C 0j = (z , µ) | yj = 0 C 1

j = (z , µ) | yj = µ

Take v0 ← µ0u0 v1 ← µ1u1

min ‖z − z‖

s.t. v0 + v1 = zµ0 + µ1 = 1(v0, µ0) ∈ C 0

(v1, µ1) ∈ C 1j

µ0, µ1 ≥ 0

Solution to example:(x∗

(−0.4010.780

separating hyperplane: ψT (z − z), where ψ ∈ ∂‖z − z‖

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Example, cont.

C 0j = (z , µ) | yj = 0 C 1

j = (z , µ) | yj = µ

Take v0 ← µ0u0 v1 ← µ1u1

min ‖z − z‖

s.t. v0 + v1 = zµ0 + µ1 = 1(v0, µ0) ∈ C 0

(v1, µ1) ∈ C 1j

µ0, µ1 ≥ 0

Solution to example:(x∗

(−0.4010.780

separating hyperplane: ψT (z − z), where ψ ∈ ∂‖z − z‖

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Example, Cont.C 0j C 1

z = (x , y)

0.198x + 0.061y = −0.032

(2x∗ + 0.5

2y∗ − 0.75

)0.198x + 0.061y ≥ −0.032

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Nonlinear Branch-and-Cut (Stubbs and Mehrotra, 1999)

Can do this at all nodes of the branch-and-bound tree

Generalize disjunctive approach from MILP

solve one convex NLP per cut

Generalizes Sherali and Adams (1990) and Lovasz and Schrijver(1991)

tighten cuts by adding semi-definite constraint

Stubbs and Mehrohtra (2002) also show how to generate convexquadratic inequalities, but computational results are not thatpromising

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Generalized Disjunctive Programming (Raman and Grossmann,

1994; Lee and Grossmann, 2000)

Consider disjunctive NLP

minimizex ,Y

∑fi + f (x)

subject to

ci (x) ≤ 0fi = γi

∨ ¬Yi

Bix = 0fi = 0

∀i ∈ I

0 ≤ x ≤ U, Ω(Y ) = true, Y ∈ true, falsep

Application: process synthesis• Yi represents presence/absence of units• Bix = 0 eliminates variables if unit absentExploit disjunctive structure• special branching ... OA/GBD algorithms

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minimizex ,Y

∑fi + f (x)

subject to

∨ ¬Yi

Bix = 0fi = 0

∀i ∈ I

Big-M formulation (notoriously bad), M > 0:

ci (x) ≤ M(1− yi )−Myi ≤ Bix ≤ Myifi = yiγi Ω(Y ) converted to linear inequalities

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minimizex ,Y

∑fi + f (x)

subject to

∨ ¬Yi

Bix = 0fi = 0

∀i ∈ I

convex hull representation ...

x = vi1 + vi0, λi1 + λi0 = 1λi1ci (vi1/λi1) ≤ 0, Bivi0 = 00 ≤ vij ≤ λijU, 0 ≤ λij ≤ 1, fi = λi1γi

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Dealing with Nonconvexities

Functional nonconvexity causesserious problems.

Branch and bound must havetrue lower bound (globalsolution)

Underestimate nonconvexfunctions. Solve relaxation.Provides lower bound.

If relaxation is not exact, thenbranch

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Dealing with Nonconvex Constraints

If nonconvexity in constraints,may need to overestimate andunderestimate the function toget a convex region

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Envelopes

f : Ω→ R

Convex Envelope (vexΩ(f )):Pointwise supremum of convexunderestimators of f over Ω.

Concave Envelope (cavΩ(f )):Pointwise infimum of concaveoverestimators of f over Ω.

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Envelopes

f : Ω→ R

Convex Envelope (vexΩ(f )):Pointwise supremum of convexunderestimators of f over Ω.

Concave Envelope (cavΩ(f )):Pointwise infimum of concaveoverestimators of f over Ω.

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Branch-and-Bound Global Optimization Methods

Under/Overestimate “simple” parts of (Factorable) Functionsindividually

Bilinear TermsTrilinear TermsFractional TermsUnivariate convex/concave terms

General nonconvex functions f (x) can be underestimated over aregion [l , u] “overpowering” the function with a quadratic functionthat is ≤ 0 on the region of interest

L(x) = f (x) +n∑

αi (li − xi )(ui − xi )

Refs: (McCormick, 1976; Adjiman et al., 1998; Tawarmalani andSahinidis, 2002)

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Branch-and-Bound Global Optimization Methods

Under/Overestimate “simple” parts of (Factorable) Functionsindividually

Bilinear TermsTrilinear TermsFractional TermsUnivariate convex/concave terms

General nonconvex functions f (x) can be underestimated over aregion [l , u] “overpowering” the function with a quadratic functionthat is ≤ 0 on the region of interest

L(x) = f (x) +n∑

αi (li − xi )(ui − xi )

Refs: (McCormick, 1976; Adjiman et al., 1998; Tawarmalani andSahinidis, 2002)

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Bilinear Terms

The convex and concave envelopes of the bilinear function xy over arectangular region

Rdef= (x , y) ∈ R2 | lx ≤ x ≤ ux , ly ≤ y ≤ uy

are given by the expressions

vexxyR(x , y) = maxlyx + lxy − lx ly , uyx + uxy − uxuycavxyR(x , y) = minuyx + lxy − lxuy , lyx + uxy − ux ly

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Worth 1000 Words?xy

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Worth 1000 Words?vexR(xy)

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Worth 1000 Words?cavR(xy)

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Summary

MINLP: Good relaxations are important

Relaxations can be improved

Statically: Better formulation/preprocessingDynamically: Cutting planes

Nonconvex MINLP:

Methods exist, again based on relaxations

Tight relaxations is an active area of research

Lots of empirical questions remain

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Part IV

Implementation and Software

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Implementation and Software for MINLP

1 Special Ordered Sets

2 Implementation & Software Issues

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Special Ordered Sets of Type 1

SOS1:∑λi = 1 & at most one λi is nonzero

Example 1: d ∈ d1, . . . , dp discrete diameters

⇔ d =∑λidi and λ1, . . . , λp is SOS1

⇔ d =∑λidi and

∑λi = 1 and λi ∈ 0, 1

. . . d is convex combination with coefficients λi

Example 2: nonlinear function c(y) of single integer

⇔ y =∑

iλi and c =∑

c(i)λi and λ1, . . . , λp is SOS1

References: (Beale, 1979; Nemhauser, G.L. and Wolsey, L.A., 1988;Williams, 1993) . . .

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Branching on SOS1

1 reference row a1 < . . . < ape.g. diameters

2 fractionality: a :=∑

3 find t : at < a ≤ at+1

4 branch: λt+1, . . . , λp = 0or λ1, . . . , λt = 0

a < at

a > at+1

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Branching on SOS1

1 reference row a1 < . . . < ape.g. diameters

aiλi3 find t : at < a ≤ at+1

4 branch: λt+1, . . . , λp = 0or λ1, . . . , λt = 0 a < a

ta > a

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SOS2:∑λi = 1 & at most two adjacent λi nonzero

Example: Approximation of nonlinear function z = z(x)

breakpoints x1 < . . . < xp

function values zi = z(xi )

piece-wise linear

x =∑λixi

z =∑λizi

λ1, . . . , λp is SOS2

. . . convex combination of two breakpoints . . .

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Example: Approximation of nonlinear function z = z(x)

breakpoints x1 < . . . < xp

function values zi = z(xi )

piece-wise linear

x =∑λixi

z =∑λizi

λ1, . . . , λp is SOS2

. . . convex combination of two breakpoints . . .

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Branching on SOS2

1 reference row a1 < . . . < ape.g. ai = xi

3 find t : at < a ≤ at+1

4 branch: λt+1, . . . , λp = 0or λ1, . . . , λt−1

tx > ax < a

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Branching on SOS2

1 reference row a1 < . . . < ape.g. ai = xi

aiλi3 find t : at < a ≤ at+1

4 branch: λt+1, . . . , λp = 0or λ1, . . . , λt−1 t

x > ax < at

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Example: Approximation of 2D function u = g(v ,w)

Triangularization of [vL, vU ]× [wL,wU ] domain

1 vL = v1 < . . . < vk = vU2 wL = w1 < . . . < wl = wU

3 function uij := g(vi ,wj)

4 λij weight of vertex (i , j)

v =∑λijvi

w =∑λijwj

u =∑λijuij

1 =∑λij is SOS3 . . .

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1 vL = v1 < . . . < vk = vU2 wL = w1 < . . . < wl = wU

v =∑λijvi

w =∑λijwj

u =∑λijuij

1 =∑λij is SOS3 . . .

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1 vL = v1 < . . . < vk = vU2 wL = w1 < . . . < wl = wU

v =∑λijvi

w =∑λijwj

u =∑λijuij

1 =∑λij is SOS3 . . .

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SOS3:∑λij = 1 & set condition holds

1 v =∑λijvi ... convex combinations

2 w =∑λijwj

3 u =∑λijuij

λ11, . . . , λkl satisfies set condition

⇔ ∃ trangle ∆ : (i , j) : λij > 0 ⊂ ∆ v

violates set condn

i.e. nonzeros in single triangle ∆

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Branching on SOS3

λ violates set condition

compute centers:v =

∑λijvi &

w =∑λijwi

find s, t such thatvs ≤ v < vs+1 &ws ≤ w < ws+1

branch on v or w

violates set condn

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Branching on SOS3

compute centers:v =

∑λijvi &

w =∑λijwi

branch on v or w

vertical branching:∑L

λij = 1∑R

λij = 1

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Branching on SOS3

compute centers:v =

∑λijvi &

w =∑λijwi

branch on v or w

horizontal branching:∑T

λij = 1∑B

λij = 1

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Extension to SOS-k

Example: electricity transmission network:

c(x) = 4x1 − x22 − 0.2 · x2x4 sin(x3)

(Martin et al., 2005) extend SOS3 to SOSk models for any k⇒ function with p variables on N grid needs Np λ’s

Alternative (Gatzke, 2005):

exploit computational graph' automatic differentiation

only need SOS2 & SOS3 ...replace nonconvex parts

piece-wise polyhedral approx.

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Extension to SOS-k

c(x) = 4x1 − x22 − 0.2 · x2x4 sin(x3)

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Extension to SOS-k

c(x) = 4x1 − x22 − 0.2 · x2x4 sin(x3)

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Extension to SOS-k

c(x) = 4x1 − x22 − 0.2 · x2x4 sin(x3)

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Software for MINLP

Outer Approximation: DICOPT++ (& AIMMS)NLP solvers: CONOPT, MINOS, SNOPTMILP solvers: CPLEX, OSL2

Branch-and-Bound Solvers: SBB & MINLPNLP solvers: CONOPT, MINOS, SNOPT & FilterSQPvariable & node selection; SOS1 & SOS2 support

Global MINLP: BARON & MINOPT underestimators & branchingCPLEX, MINOS, SNOPT, OSL

Online Tools: MINLP World, MacMINLP & NEOS MINLP Worldwww.gamsworld.org/minlp/

NEOS server www-neos.mcs.anl.gov/

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COIN-OR

http://www.coin-or.org

COmputational INfrastructure for Operations Research

A library of (interoperable) software tools for optimization

A development platform for open source projects in the ORcommunity

Possibly Relevant Modules:

OSI: Open Solver InterfaceCGL: Cut Generation LibraryCLP: Coin Linear Programming ToolkitCBC: Coin Branch and CutIPOPT: Interior Point OPTimizer for NLPNLPAPI: NonLinear Programming API

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MINLP with COIN-OR

New implementation of LP/NLP based BB

MIP branch-and-cut: CBC & CGL

NLPs: IPOPT interior point ... OK for NLP(yi )

New hybrid method:

solve more NLPs at non-integer yi⇒ better outer approximationallow complete MIP at some nodes⇒ generate new integer assignment

... faster than DICOPT++, SBB

simplifies to OA and BB at extremes ... less efficient

... see Bonami et al. (2005) ... coming in 2006.

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Conclusions

MINLP rich modeling paradigm most popular solver on NEOS

Algorithms for MINLP: Branch-and-bound (branch-and-cut) Outer approximation et al.

“MINLP solvers lag 15 years behind MIP solvers”

⇒ many research opportunities!!!

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Conclusions

MINLP rich modeling paradigm most popular solver on NEOS

Algorithms for MINLP: Branch-and-bound (branch-and-cut) Outer approximation et al.

“MINLP solvers lag 15 years behind MIP solvers”

⇒ many research opportunities!!!

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Part V

References

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