A useful algebraic representation of disjunctive convex sets

A useful algebraic representation of disjunctive convex sets using the perspective function Kevin Furman 1 Nicolas Sawaya 2 Ignacio Grossmann 3 1 Exxon. Mobil Research & Engineering 2 Exxon. Mobil Gas & Power Marketing 3 Carnegie Mellon University MINLP 2014, CMU

Motivation • Disjunctive convex sets arise in numerous applications – – – process synthesis (heat exchanger networks and reactor networks) engineering design (truss structures and feed location in distillation columns) planning of process networks optimal positioning of products financial planning scheduling of batch and continuous multiproduct batch plants • Tight representation of the disjunctive convex set is sought, either to be explicitly used in the reformulation or to be implicitly exploited through the generation of cutting planes • Tightest such representation involves the characterization of the convex hull of the disjunctive convex set – In the most general case, can be explicitly expressed through the use of the perspective function in higher dimensional space (extended formulation) • Number of challenges in using extended formulation in computation including nondifferentiability of perspective function at λ=0. • Goal of this work is to propose an algebraic representation of a fairly large class of disjunctive convex sets using the perspective function that addresses (some! of) these computational challenges.

, Background Convex disjunctive sets Perspective Function

Convex Hull of Disjunctive Convex Sets Ceria & Soares [1999] characterize the closure of the convex hull of the set F using the perspective function (see also Stubbs & Mehrotra [1999] and Jereslow [1987]) Major computational challenge: How do we take care of non-differentiability at λ=0?

What to do? Stubbs & Mehrotra (1999) • Do not discuss implementation of perspective relaxation in their paper. • Report numerical convergence issues when using algorithm for continuously differentiable optimization within the context of generating disjunctive cutting planes Ceria & Soares (1999) • Propose a log-barrier approach o Requires the solution of many convex programs. o Termination criteria to guarantee equivalence with original problem is not straightforward. o No readily implemented version of the algorithm is available. What to do? Three major strategies have been used to tackle non-differentiability: 1. If disjunctive sets have particular structure: generate explicit formulation that avoids problem 2. Generate (specialized or general) cutting planes that implicitly approximate convex hull 3. If disjunctive sets have no particular structure: generate explicit formulation and add small ε-term to avoid division by 0

Strategy 1: Explicit formulations for special disjunctive sets Gunluk, Linderoth (2009) • • Indicator induced {0, 1} MINLP Avoid dealing directly with non-differentiability at 0 by virtue of the description of the convex hull in original space, which is represented as (rotated) second-order cone constraints; can solve as SOCP. Good computational results, but limited to special disjunctive sets (union of point & convex region)

Strategy 2: Implicitly approximate convex hull by using cutting planes 1) For MINLPs with semi-continuous variables Frangioni, Gentile (2006, 2009) Perspective reformulation (tighter relaxation) • • Avoid dealing directly with perspective function by iteratively approximating it using linear “perspective cuts” Good computational results but limited to special problems (perspective cut equivalent to generating outerapproximation to the convex hull of union of point and disjunctive convex set) 2) For (convex) split disjunctions (i. e. split disjunctions whose terms are convex sets) Zhu & Kuno (2006) • • Avoid dealing with perspective function by generating cuts from linear outer-approximation to nonlinear relaxation Cuts can be weak -> Limited computational success on larger problems Kilinc, Linderoth & Luedtke (2010) • • Avoid dealing with perspective function by generating cuts refined by constraint generation from linear outer-approximation to nonlinear relaxation Cuts are strong (equal to Stubbs & Mehrotra at limit), but constraint generation can have slow convergence properties Bonami (2011) • • • Generates lift & project cuts in 2 steps: 1) nonlinear separation problem in original space 2) linear outer-approximation Extends Balas & Perregaard (2002) insight in generating lift & project cuts in original space to nonlinear case Good computational results, but procedure limited only to split disjunctions

Strategy 3: ε-approximations Lee & Grossmann (2000) by Replace 1. The divisibility by 0 problem is avoided. 2. The new constraints are equivalent to the original constraints as ε → 0. 3. The LHS of the new constraints are convex. Problem: When the new constraints are infeasible. and Sawaya & Grossmann (2007) Replace by: 1. The divisibility by 0 problem is avoided. 2. The new constraints are equivalent to the original constraints as ε → 0. 3. The LHS of the new constraints are convex. the new constraints are feasible. 4. When and Problem: When , approximation is not exact for finite ε. In rare cases, this may lead to different optimal solution unless ε is exceptionally small, which causes numerical difficulties.

Our Proposal* Furman, Sawaya & Grossmann (to be submitted) by: Replace 1. The divisibility by 0 problem is avoided. 2. The new constraints are equivalent to the original constraints as ε → 0. 3. The LHS of the new constraints are convex. 4. When and the new constraints are feasible. 5. The new constraints are equivalent to the original constraints at yjk = 0 and at yjk = 1 regardless of the value of ε. 6. Established criteria to guarantee tightness of formulation relative to big-M. *Appeared in Sawaya’s thesis (2006) based on personal communication from Furman (2006); also presented at ISMP (2009)

Some theoretical results Define the disjunctive convex set F proj(x) (eps-MIP F( )) = F -approximation is convex Define the set: eps-rel F( ) Feasible region of eps-rel F( ) is compact set proj(x) (eps-MIP F( )) is relaxation of F proj(x) (eps-MIP F( )) equivalent to conv(F) as ->0 Define the set: eps-MIP F( ) Define the set: proj(x) (eps-MIP F( ))

Application of theory to Generalized Disjunctive Programming (GDP). What is a GDP? Raman R. and Grossmann I. E. (1994) (GDP) Objective function Common constraints Disjunctive constraints Logical OR operator Boolean variables

Transformation of GDP to DP Sawaya & Grossmann (2012) / Ruiz & Grosmmann (2012) Nonlinear GDP Nonlinear DP Essentially, GDP problems are {0, 1} indicator-induced mixed-integer programs

BIG-M MINLP REFORMULATION for GDP Lee S. and Grossmann I. E. (2000) Big-M parameters (BIG-M)

HULL MINLP REFORMULATION for GDP Lee S. and Grossmann I. E. (2000) (CH) Replace with:

GDP EXAMPLE 1: SYNTHESIS OF PROCESS NETWORK Problem Statement: Duran & Grossmann (1986) - Synthesis of process network. · Superstructure involves possible selection of processes. · Every process has a fixed cost associated with it. - Objective is to obtain network that minimizes cost. x 14 x 2 x 1 A Raw Material x 4 1 OR 2 x 19 x 3 x 12 x 5 x 11 4 OR B x 15 x 6 x 8 5 3 x 13 E x 21 6 x 20 OR 7 x 25 x 16 x 17 x 9 8 C x 10 x 24 x 22 x 23 x 18 F Products D

GDP EXAMPLE 2: RETROFIT & SYNTHESIS PLANNING PROBLEM Problem Statement: Sawaya & Grossmann (2006) - Simultaneous Retrofit & Synthesis of Plant - Retrofit: Redesign of existing plant. · Improvements such as higher yield, increased capacity, energy reduction. - Objective is to identify modifications that maximize economic potential, given time horizon and limited capital investments. Increase capacity A Process 1 D B Process 2 E No modifications C Process 3 Increase conversion and capacity Plant-wide energy reduction

GDP EXAMPLE 3: CONSTRAINED LAYOUT Problem statement: Sawaya &Grossmann (2006) – Problem consists of placing non-overlapping units represented by rectangles within the confines of certain designated areas formulated as circular nonlinear constraints, such that the cost of connecting these units is minimized. y 1 3 2 x

COMPUTATIONAL EXPERIMENTS • Computations performed using nonlinear branch-andbound (GAMS/SBB) • 5 hour time limit on 2. 4 GHz 8 GB RAM Linux PC • Time is reported in seconds • 52 total instances – 24 Synthesis – 22 Retrofit Synthesis – 6 Constrained Layout • Used ε values ranging from 10 -10 to 0. 99

ROOT RELAXATION Perspective function used in hull relaxation • For ε of 10 -10, 12 of 22 retrofit instances experienced numerical errors in solving the root relaxation • Clear increase in relaxation gap as ε increases for synthesis and retrofit – Big-M is substantially worse • Constrained layout shows no difference in relaxation between ε and big-M

SOLVABILITY OF INSTANCES Constrained Layout (6 instances) Retrofit (22 instances) • • Overall (52 instances) Using a value for ε of 10 -10 has an increased instance of failures due to numerical difficulties As ε increases and the big-M formulations tend to time-out more often – • Synthesis (24 instances) most likely due to a weaker relaxation Wide range of values for ε give good performance (from 10 -4 to 10 -1 )

CUMULATIVE RESULTS FOR SHARED INSTANCES • Considering instances in which all values of ε and the big-M formulations were solvable (within the time limit) • Reporting cumulative number of nodes and amount of time • As implied by relaxation results, big-M for constrained layout is faster, but substantially slower for the other two instance sets • For synthesis and retrofit instance sets, ε in the range of 0. 1 to 0. 001 appears to be best

CONCLUSIONS • An explicit algebraic representation of disjunctive convex sets is presented using the perspective function • • • This reformulation avoids the implementation problems of previous attempts dealing with perspective function formulations of nonlinear disjunctive programs • • • The proposed reformulation is general and does not depend on the particular structure of the disjunctions This facilitates implementation via general purpose algebraic modeling languages and/or using general purpose solvers Remains exact for any value of ε between 0 and 1 Wide range of values for ε (from 10 -4 to 10 -1 ) give good numerical performance This reformulation can also be used within a cutting plane scheme to generate strong cuts in original space (although separation problem in extended space)

QUESTIONS?

GDP MODEL FOR SYNTHESIS OF PROCESS NETWORK Objective function Minimize cost Common constraints Mass Balances Common constraints Design Specifications Disjunctive constraints Process Scenarios Nonlinearities Logic constraints Connect disjunctions

GDP MODEL FOR RETROFIT & SYNTHESIS PLANNING PROBLEM Objective function Maximize economic potential Common constraints Mass balances Disjunctive constraints Conversion/Capacity scenarios Disjunctive constraints Energy reduction scenarios Disjunctive constraints Process Scenarios Nonlinearities Common constraint Investment limit Logic constraints Connect disjunctions

GDP MODEL FOR CONSTRAINED LAYOUT Objective function Minimize Cost Common constraints Distance Constraints between units Disjunctive constraints No overlap of units Disjunctive constraints Circular Regions Nonlinearities

NUMERICAL EXAMPLES MINLP Problems obtained from CMU/IBM test library http: //egon. cheme. cmu. edu/ibm/page. htm - Test problems were gathered from different sources or created by the authors. - Mainly applications from OR and Chemical Engineering. - Over 150 (convex) problems from 9 different classes, in GAMS and AMPL format, including: - Synthesis of process networks - Simultaneous retrofit and synthesis of process networks - Design of Multi-product batch plants - Safety layout - Constrained layout - Farm layout - Water Networks