OPTIMIZING A PARAMETRIC LINEAR FUNCTION OVER A NON-COMPACT REAL - PowerPoint PPT Presentation

OPTIMIZING A PARAMETRIC LINEAR FUNCTION OVER A NON-COMPACT REAL VARIETY Feng Guo 1 Mohab Safey El Din 2 Chu Wang 3 Lihong Zhi 3 1 School of Mathematical Sciences, Dalian University of Technology, China 2 Sorbonne Universités, UPMC, Univ Paris 6, France INRIA Paris-Rocquencourt, POLSYS Project-Team 3 Key Lab of Mathematics Mechanization, Academy of Mathematics and System Sciences, China ISSAC’2015, Bath, July 6-10

Problem Statements Let h 1 , . . . , h p be polynomials in R [ X ] which define the algebraic variety V = { x ∈ C n | h 1 ( x ) = · · · = h p ( x ) = 0 } .

Problem Statements Let h 1 , . . . , h p be polynomials in R [ X ] which define the algebraic variety V = { x ∈ C n | h 1 ( x ) = · · · = h p ( x ) = 0 } . Consider the following optimization problem c T x = c 1 x 1 + · · · + c n x n , c ∗ 0 := sup x ∈V∩ R n where c = ( c 1 , . . . , c n ) denotes the coefficient vector.

Problem Statements Let h 1 , . . . , h p be polynomials in R [ X ] which define the algebraic variety V = { x ∈ C n | h 1 ( x ) = · · · = h p ( x ) = 0 } . Consider the following optimization problem c T x = c 1 x 1 + · · · + c n x n , c ∗ 0 := sup x ∈V∩ R n where c = ( c 1 , . . . , c n ) denotes the coefficient vector. Tarski-Seidenberg’s theorem on quantifier elimination ensures that optimal value function c ∗ 0 is a semialgebraic function.

Problem Statements Let h 1 , . . . , h p be polynomials in R [ X ] which define the algebraic variety V = { x ∈ C n | h 1 ( x ) = · · · = h p ( x ) = 0 } . Consider the following optimization problem c T x = c 1 x 1 + · · · + c n x n , c ∗ 0 := sup x ∈V∩ R n where c = ( c 1 , . . . , c n ) denotes the coefficient vector. Tarski-Seidenberg’s theorem on quantifier elimination ensures that optimal value function c ∗ 0 is a semialgebraic function. The Problem ◮ How to compute a polynomial Φ ∈ R [ c 0 , c ] s.t. c ∗ 0 can be obtained by solving Φ( c 0 , γ ) = 0 for a generic γ ∈ R n ?

Problem Statements Let h 1 , . . . , h p be polynomials in R [ X ] which define the algebraic variety V = { x ∈ C n | h 1 ( x ) = · · · = h p ( x ) = 0 } . Consider the following optimization problem c T x = c 1 x 1 + · · · + c n x n , c ∗ 0 := sup x ∈V∩ R n where c = ( c 1 , . . . , c n ) denotes the coefficient vector. Tarski-Seidenberg’s theorem on quantifier elimination ensures that optimal value function c ∗ 0 is a semialgebraic function. The Problem ◮ How to compute a polynomial Φ ∈ R [ c 0 , c ] s.t. c ∗ 0 can be obtained by solving Φ( c 0 , γ ) = 0 for a generic γ ∈ R n ? ◮ Can we compute a polynomial family { Φ i } ∈ R [ c 0 , c ] , s.t. ∀ γ ∈ R n , there exists k Φ k ( c 0 , γ ) �≡ 0 , Φ k ( c ∗ 0 , γ ) = 0?

State of the Art Previous work on the problem: ◮ CAD can be used to describe the optimal value function by a sequence of polynomials of degree doubly exponential in n . [Brown, Collins, Hong, McCallum among many others]

State of the Art Previous work on the problem: ◮ CAD can be used to describe the optimal value function by a sequence of polynomials of degree doubly exponential in n . [Brown, Collins, Hong, McCallum among many others] ◮ When V is irreducible, compact and smooth in R n , Rostalski and Sturmfels give such a polynomial Φ for generic parameter’s value γ by computing dual variety [Rostalski, Sturmfels]. Dual variety has degree singly exponential in n .

State of the Art Previous work on the problem: ◮ CAD can be used to describe the optimal value function by a sequence of polynomials of degree doubly exponential in n . [Brown, Collins, Hong, McCallum among many others] ◮ When V is irreducible, compact and smooth in R n , Rostalski and Sturmfels give such a polynomial Φ for generic parameter’s value γ by computing dual variety [Rostalski, Sturmfels]. Dual variety has degree singly exponential in n . ◮ For the specialized optimization problem, the algorithm based on modified polar varieties [Greuet, Guo, Safey El Din, Zhi], [Greuet, Safey El Din] allows us to compute a polynomial of degree singly exponential in n whose roots contain the maximal value. It works for noncompact cases.

Our Contributions We generalize the results of Rostalski and Sturmfels and have the following conclusions: ◮ When V is nonsmooth and compact in R n , dual varieties of regular locus and singular locus give such a polynomial Φ for generic parameter’s value γ .

Our Contributions We generalize the results of Rostalski and Sturmfels and have the following conclusions: ◮ When V is nonsmooth and compact in R n , dual varieties of regular locus and singular locus give such a polynomial Φ for generic parameter’s value γ . ◮ When V is smooth and noncompact in R n , dual variety V ∗ gives such a polynomial Φ for generic parameter’s value γ .

Our Contributions We generalize the results of Rostalski and Sturmfels and have the following conclusions: ◮ When V is nonsmooth and compact in R n , dual varieties of regular locus and singular locus give such a polynomial Φ for generic parameter’s value γ . ◮ When V is smooth and noncompact in R n , dual variety V ∗ gives such a polynomial Φ for generic parameter’s value γ . We compute finitely many pairs of polynomials (Φ i , Z i ) where Φ i ∈ Q [ c 0 , c ] and Z i ∈ Q [ c ] such that ◮ for each γ , there exists k such that γ �∈ V ( Z k ) and Φ k ( c 0 , γ ) �≡ 0 ; ◮ if c ∗ 0 is finite for γ , Φ k ( c ∗ 0 , γ ) = 0 .

Main Results for Smooth and Noncompact Case Let V ∗ ⊂ P n be the dual variety to the projective closure of V and C h the closure of the convex hull of V ∩ R n . We extend the result of [Rostalski, Sturmfels] and have the following conclusions: Theorem Suppose that V is equidimensional and smooth, then ( − c ∗ 0 : γ 1 : · · · : γ n ) ∈ V ∗ for every γ such that c ∗ 0 is finite.

Main Results for Smooth and Noncompact Case Let V ∗ ⊂ P n be the dual variety to the projective closure of V and C h the closure of the convex hull of V ∩ R n . We extend the result of [Rostalski, Sturmfels] and have the following conclusions: Theorem Suppose that V is equidimensional and smooth, then ( − c ∗ 0 : γ 1 : · · · : γ n ) ∈ V ∗ for every γ such that c ∗ 0 is finite. Theorem When V is irreducible, smooth and C h contains no lines, we have ◮ V ∗ is an irreducible hypersurface, ◮ its defining polynomial is Φ( − c 0 , c 1 , . . . , c n ) , where Φ( c ∗ 0 , γ ) = 0 for each γ ∈ R n .

Bad Parameters’ Values Cases Φ = Φ 0 ( c 1 , . . . , c n ) c m 0 + Φ 1 ( c 1 , . . . , c n ) c m − 1 + · · · + Φ m ( c 1 , . . . , c n ) 0 Example We consider the optimization problem: s.t. x ∈ V ∩ R 4 , sup c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x 4 1 x 2 x 3 ) 2 ) . The dual variety V ∗ is defined where V = V ( x 4 − ( x 1 + x 2 1 x 2 2 + x 4 by Φ :=1073741824 c 12 0 c 4 2 c 2 4 + 268435456 c 11 0 c 2 1 c 4 2 c 4 − 134217728 c 11 0 c 2 2 c 2 3 c 3 4 − 33554432 c 10 0 c 2 1 c 2 2 c 2 3 c 2 4 + · · · + 520093696 c 9 0 c 1 c 3 2 c 2 3 c 3 4 . ◮ When γ ∈ V ( c 2 c 4 , c 3 c 4 , c 3 c 2 c 1 ) , Φ( c 0 , γ ) ≡ 0 . ◮ Φ( c 0 , 0 , 0 , 0 , − 1) ≡ 0 which gives no info on c ∗ 0 .

Algorithm for Non-parametric Optimization Construct a one dimensional subvariety C ⊆ V such that x ∈V∩ R n f ( x ) = sup x ∈C∩ R n f ( x ) . sup

Algorithm for Non-parametric Optimization Construct a one dimensional subvariety C ⊆ V such that x ∈V∩ R n f ( x ) = sup x ∈C∩ R n f ( x ) . sup Algorithm: [Greuet and Safey El Din’14] Input: f ( X ) := γ T X , � h 1 , . . . , h p � ∈ Q [ X ] and γ ∈ Q n

Algorithm for Non-parametric Optimization Construct a one dimensional subvariety C ⊆ V such that x ∈V∩ R n f ( x ) = sup x ∈C∩ R n f ( x ) . sup Algorithm: [Greuet and Safey El Din’14] Input: f ( X ) := γ T X , � h 1 , . . . , h p � ∈ Q [ X ] and γ ∈ Q n ◮ Construct C := W \ Crit ( f, V ) .

Algorithm for Non-parametric Optimization Construct a one dimensional subvariety C ⊆ V such that x ∈V∩ R n f ( x ) = sup x ∈C∩ R n f ( x ) . sup Algorithm: [Greuet and Safey El Din’14] Input: f ( X ) := γ T X , � h 1 , . . . , h p � ∈ Q [ X ] and γ ∈ Q n ◮ Construct C := W \ Crit ( f, V ) . ◮ S 1 : a set of sample points in each connected component of V ∩ R n − → isolated local extrema ∈ f ( S 1 ) .

Algorithm for Non-parametric Optimization Construct a one dimensional subvariety C ⊆ V such that x ∈V∩ R n f ( x ) = sup x ∈C∩ R n f ( x ) . sup Algorithm: [Greuet and Safey El Din’14] Input: f ( X ) := γ T X , � h 1 , . . . , h p � ∈ Q [ X ] and γ ∈ Q n ◮ Construct C := W \ Crit ( f, V ) . ◮ S 1 : a set of sample points in each connected component of V ∩ R n − → isolated local extrema ∈ f ( S 1 ) . ◮ S 2 := C ∩ Crit ( f, V ) − → critical values ∈ f ( S 2 ) .