Using mixed volume theory to compute the convex hull volume for - PowerPoint PPT Presentation

Emily Speakman Gennadiy Averkov Using mixed volume theory to compute the convex hull volume for trilinear monomials 23 rd Combinatorial Optimization Workshop, Aussois January 9, 2019 Institute of Mathematical Optimization, Otto-von-Guericke-University, Magdeburg, Germany.

Talk Outline • Using volume to compare relaxations for mixed-integer nonlinear optimization • The convex hull of the graph of a trilinear monomial over a box • Use techniques from mixed volume theory to obtain an alternative proof 1 Speakman & Averkov // Mixed Volume

Global optimization of non-convex functions is hard! x ∈ R n , z ∈ Z m { f ( x , z ) : ( x , z ) ∈ F} min • Global optimization of a mixed integer non-linear optimization (MINLO) problem • Not necessarily convex sets/functions 2 Speakman & Averkov // Mixed Volume

Global optimization of non-convex functions is hard! x ∈ R n , z ∈ Z m { f ( x , z ) : ( x , z ) ∈ F} min • Global optimization of a mixed integer non-linear optimization (MINLO) problem • Not necessarily convex sets/functions Software: Baron , Couenne , Scip , Antigone ... 2 Speakman & Averkov // Mixed Volume

Global optimization of non-convex functions is hard! x ∈ R n , z ∈ Z m { f ( x , z ) : ( x , z ) ∈ F} min • Global optimization of a mixed integer non-linear optimization (MINLO) problem • Not necessarily convex sets/functions Software: Baron , Couenne , Scip , Antigone ... • Each of these perform some variation on the algorithm known as Spatial Branch-and-Bound (sBB) 2 Speakman & Averkov // Mixed Volume

Global optimization of non-convex functions is hard! x ∈ R n , z ∈ Z m { f ( x , z ) : ( x , z ) ∈ F} min • Global optimization of a mixed integer non-linear optimization (MINLO) problem • Not necessarily convex sets/functions Software: Baron , Couenne , Scip , Antigone ... • Each of these perform some variation on the algorithm known as Spatial Branch-and-Bound (sBB) • sBB has a daunting complexity 2 Speakman & Averkov // Mixed Volume

Global optimization of non-convex functions is hard! x ∈ R n , z ∈ Z m { f ( x , z ) : ( x , z ) ∈ F} min • Global optimization of a mixed integer non-linear optimization (MINLO) problem • Not necessarily convex sets/functions Software: Baron , Couenne , Scip , Antigone ... • Each of these perform some variation on the algorithm known as Spatial Branch-and-Bound (sBB) • sBB has a daunting complexity • How can we effectively tune/engineer this software? • experimentally • mathematically 2 Speakman & Averkov // Mixed Volume

Key algorithm: sBB To find optimal solutions we use Spatial Branch- and-Bound : • Create sub-problems by branching on individual variables • Generate convex relaxations of the graph of the function at each node 3 Speakman & Averkov // Mixed Volume

The choice of convexification method is important Computational tractability of sBB depends on the quality of the convexifications we use. We want both: • Tight relaxations (good bounds) • Simple algebraic representations of feasible regions (solve quickly) We have a trade off: 4 Speakman & Averkov // Mixed Volume

We compare the tightness of convexifications via n -dimensional volume • Allows us to quantify the difference between formulations analytically • The optimal solution could occur anywhere in the feasible region, and therefore the volume measure corresponds to a uniform distribution on the location of the optimal solution • Volume was introduced as a means of comparing formulations by Lee and Morris (1994) • Recent survey on volumetric comparison of polyhedral relaxations for optimization Lee, Skipper, Speakman (2018) 5 Speakman & Averkov // Mixed Volume

Trilinear monomials: some notation Assume we have a monomial of the form: f = x 1 x 2 x 3 . 6 Speakman & Averkov // Mixed Volume

Trilinear monomials: some notation Assume we have a monomial of the form: f = x 1 x 2 x 3 . • x i ∈ [ a i , b i ] , where 0 ≤ a i < b i , for i = 1 , 2 , 3 6 Speakman & Averkov // Mixed Volume

Trilinear monomials: some notation Assume we have a monomial of the form: f = x 1 x 2 x 3 . • x i ∈ [ a i , b i ] , where 0 ≤ a i < b i , for i = 1 , 2 , 3 • Label the variables x 1 , x 2 and x 3 such that: a 1 ≤ a 2 ≤ a 3 ( Ω ) b 1 b 2 b 3 6 Speakman & Averkov // Mixed Volume

Trilinear monomials: some notation Assume we have a monomial of the form: f = x 1 x 2 x 3 . • x i ∈ [ a i , b i ] , where 0 ≤ a i < b i , for i = 1 , 2 , 3 • Label the variables x 1 , x 2 and x 3 such that: a 1 ≤ a 2 ≤ a 3 ( Ω ) b 1 b 2 b 3 The convex hull of the graph f = x 1 x 2 x 3 (on the domain x i ∈ [ a i , b i ] ) is polyhedral, we refer to this polytope as P H . 6 Speakman & Averkov // Mixed Volume

Trilinear monomials: some notation Assume we have a monomial of the form: f = x 1 x 2 x 3 . • x i ∈ [ a i , b i ] , where 0 ≤ a i < b i , for i = 1 , 2 , 3 • Label the variables x 1 , x 2 and x 3 such that: a 1 ≤ a 2 ≤ a 3 ( Ω ) b 1 b 2 b 3 The convex hull of the graph f = x 1 x 2 x 3 (on the domain x i ∈ [ a i , b i ] ) is polyhedral, we refer to this polytope as P H . • The facet description was given by Meyer and Floudas (2004) • There are alternative convexifications for trilinear monomials (based on the well-know McCormick inequalities) • S. and Lee (2017) consider these alternatives and compute their volumes • Here we focus on the convex hull i.e. P H 6 Speakman & Averkov // Mixed Volume

Convex hull of the graph f = x 1 x 2 x 3 over a box • The extreme points of P H are the 8 points that correspond to the 2 3 = 8 choices of each x -variable at its upper or lower bound:         b 1 a 2 a 3 a 1 a 2 a 3 a 1 a 2 b 3 a 1 b 2 a 3 b 1 a 1 a 1 a 1 v 1 :=  v 2 :=  v 3 :=  v 4 :=                 a 2 a 2 a 2 b 2      a 3 a 3 b 3 a 3         a 1 b 2 b 3 b 1 b 2 b 3 b 1 b 2 a 3 b 1 a 2 b 3 a 1 b 1 b 1 b 1 v 5 :=  v 6 :=  v 7 :=  v 8 :=                 b 2 b 2 b 2 a 2      b 3 b 3 a 3 b 3 7 Speakman & Averkov // Mixed Volume

Convex hull of the graph f = x 1 x 2 x 3 over a box • The extreme points of P H are the 8 points that correspond to the 2 3 = 8 choices of each x -variable at its upper or lower bound:         b 1 a 2 a 3 a 1 a 2 a 3 a 1 a 2 b 3 a 1 b 2 a 3 b 1 a 1 a 1 a 1 v 1 :=  v 2 :=  v 3 :=  v 4 :=                 a 2 a 2 a 2 b 2      a 3 a 3 b 3 a 3         a 1 b 2 b 3 b 1 b 2 b 3 b 1 b 2 a 3 b 1 a 2 b 3 a 1 b 1 b 1 b 1 v 5 :=  v 6 :=  v 7 :=  v 8 :=                 b 2 b 2 b 2 a 2      b 3 b 3 a 3 b 3 • Meyer and Floudas (2004) completely characterized the facets of P H • They did this for our special case (non-negative) and also in general 7 Speakman & Averkov // Mixed Volume

Facet Description P H (Meyer and Floudas, 2004) f − a 2 a 3 x 1 − a 1 a 3 x 2 − a 1 a 2 x 3 + 2 a 1 a 2 a 3 ≥ 0 f − b 2 b 3 x 1 − b 1 b 3 x 2 − b 1 b 2 x 3 + 2 b 1 b 2 b 3 ≥ 0 f − a 2 b 3 x 1 − a 1 b 3 x 2 − b 1 a 2 x 3 + a 1 a 2 b 3 + b 1 a 2 b 3 ≥ 0 f − b 2 a 3 x 1 − b 1 a 3 x 2 − a 1 b 2 x 3 + b 1 b 2 a 3 + a 1 b 2 a 3 ≥ 0 ( ) η 1 a 1 η 1 x 1 − b 1 a 3 x 2 − b 1 a 2 x 3 + + b 1 b 2 a 3 + b 1 a 2 b 3 − a 1 b 2 b 3 f − ≥ 0 b 1 − a 1 b 1 − a 1 ( ) η 2 b 1 η 2 x 1 − a 1 b 3 x 2 − a 1 b 2 x 3 + + a 1 a 2 b 3 + a 1 b 2 a 3 − b 1 a 2 a 3 f − ≥ 0 a 1 − b 1 a 1 − b 1 − f + a 2 a 3 x 1 + b 1 a 3 x 2 + b 1 b 2 x 3 − b 1 b 2 a 3 − b 1 a 2 a 3 ≥ 0 − f + b 2 a 3 x 1 + a 1 a 3 x 2 + b 1 b 2 x 3 − b 1 b 2 a 3 − a 1 b 2 a 3 ≥ 0 − f + a 2 a 3 x 1 + b 1 b 3 x 2 + b 1 a 2 x 3 − b 1 a 2 b 3 − b 1 a 2 a 3 ≥ 0 − f + b 2 b 3 x 1 + a 1 a 3 x 2 + a 1 b 2 x 3 − a 1 b 2 b 3 − a 1 b 2 a 3 ≥ 0 − f + a 2 b 3 x 1 + b 1 b 3 x 2 + a 1 a 2 x 3 − b 1 a 2 b 3 − a 1 a 2 b 3 ≥ 0 − f + b 2 b 3 x 1 + a 1 b 3 x 2 + a 1 a 2 x 3 − a 1 b 2 b 3 − a 1 a 2 b 3 ≥ 0 a i ≤ x i ≤ b i , i = 1 .. 3 where η 1 = b 1 b 2 a 3 − a 1 b 2 b 3 − b 1 a 2 a 3 + b 1 a 2 b 3 and η 2 = a 1 a 2 b 3 − b 1 a 2 a 3 − a 1 b 2 b 3 + a 1 b 2 a 3 . 8 Speakman & Averkov // Mixed Volume

Volume of P H Theorem (S. and Lee, 2017) Under Ω , the volume of P H is given by: 1 24 ( b 1 − a 1 )( b 2 − a 2 )( b 3 − a 3 ) × ( ) b 1 ( 5 b 2 b 3 − a 2 b 3 − b 2 a 3 − 3 a 2 a 3 ) + a 1 ( 5 a 2 a 3 − b 2 a 3 − a 2 b 3 − 3 b 2 b 3 ) . 9 Speakman & Averkov // Mixed Volume

Our contribution • We provide an alternative way to obtain the formula for the convex hull volume • Observe a special structure in the convex hull polytope • Allows us to use theory from so-called Mixed Volumes 10 Speakman & Averkov // Mixed Volume