Minimizing Cubic and Homogeneous Polynomials Over Integer Points in the Plane

Transcription

1 Minimizing Cubic and Homogeneous Polynomials Over Integer Points in the Plane Robert Hildebrand IFOR - ETH Zürich Joint work with Alberto Del Pia, Robert Weismantel and Kevin Zemmer Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

2 Introduction Consider the problem min{f(x) : x P Z n }, (1) where P is the rational polyhedron P = {x R n : Ax b} with A Z m n, b Z m, f Z[x] has degree d. Complexity: We want time-complexity when input is binary encoding of A, b and the coefficients of f. Fixed: Dimension n, degree d of f. Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

3 Hardness Results 1 NP-Hard in 2 variables degree 4, P bounded [DHKW, 2005]. min{(x 2 ay c) 2 : 1 x c 1, 1 a b y (c 1)2 a b } Implies AN1 problem - given three positive integers a, b, c, determine if there exist x Z such that x 2 a (mod b) with x < c. 2 NP-Hard in 3 variables degree 4, P bounded, f homogeneous. 3 "Intractable" in 2 variables, degree 4, P unbounded, f homogeneous. min{ ( x 2 Ny 2) 2 : (x, y) Z 2 1 }, N = 5 2k+1 (2) Optimal objective value is 1. Minimal size solution satisfies Negative Pell Equation x 2 Ny 2 = 1. Lagarias (1980) - minimal solution has binary encoding size Ω(5 k ). Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

4 Positive Results 1 FPTAS for non-negative maximization in fixed dimension (DeLoera, Hemmeke, Köppe, Weismantel 2011) 2 Quadratic polynomials in dimension 2 (Del-Pia and Weismantel 2013) Theorem (Del Pia, H., Weismantel, Zemmer) The problem min{f(x) : x P Z 2 } can be solved in polynomial time in the following cases 1 f is fixed degree, homogeneous, P is bounded, 2 f is cubic, P is bounded, 3 f is cubic, homogeneous, P is unbounded. Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

5 Positive Results 1 FPTAS for non-negative maximization in fixed dimension (DeLoera, Hemmeke, Köppe, Weismantel 2011) 2 Quadratic polynomials in dimension 2 (Del-Pia and Weismantel 2013) Theorem (Del Pia, H., Weismantel, Zemmer) The problem min{f(x) : x P Z 2 } can be solved in polynomial time in the following cases 1 f is fixed degree, homogeneous, P is bounded, 2 f is cubic, P is bounded*, 3 f is cubic, homogeneous, P is unbounded. *COMPLETES COMPLEXITY CLASSIFICATION FOR BOUNDED P IN DIMENSION 2! Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

6 Quadratic Case Del Pia and Weismantel showed that the quadratic case can be solved in polynomial time. f(x, y) x 2 + y 2 f(x, y) x 2 y 2 f(x, y) x 2 y 2 Convex: Apply Khachiyan-Porkolab 2010 test feasibility + binary search. Concave: implies solution on vertex of integer hull P I. Enumerate vertices using CKHM Divide problem into quasi-convex and quasi-concave regions. Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

7 Homogeneous Case Goal: Quasiconvex/Quasiconcave division Zeros occur on lines. Function no longer quasiconcave/quasiconvex in positive/negative regions. Domain can still be divided into quasiconcave/quasiconvex regions. Numerically approximate regions. (x + y)(x y)(x 2 + y 2 ) 4 Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

8 Homogeneous Case Goal: Quasiconvex/Quasiconcave division Zeros occur on lines. Function no longer quasiconcave/quasiconvex in positive/negative regions. Domain can still be divided into quasiconcave/quasiconvex regions. Numerically approximate regions. (x + y)(x y)(x 2 + y 2 ) 4 Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

9 Bordered Hessian, Homogeneous, Volume Argument 1 Bordered Hessian [ ] 0 h T D h = det h 2 h = d d 1 h(x) det( 2 h(x)) 2 D h < 0 quasiconvex while D h > 0 quasiconcave. 3 D h 0 h(x) = (c T x) d (Hemmer, 1995) 4 D h is a homogeneous polynomial zeros occur on lines. Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

10 Bordered Hessian, Homogeneous, Volume Argument 1 Bordered Hessian [ ] 0 h T D h = det h 2 h = d d 1 h(x) det( 2 h(x)) 2 D h < 0 quasiconvex while D h > 0 quasiconcave. 3 D h 0 h(x) = (c T x) d (Hemmer, 1995) 4 D h is a homogeneous polynomial zeros occur on lines. x 2 = R x 2 = R α 1 α 2 α 3 ** 0 ** β 1 β 2 β 3 1 Compute zeros of D h (x 1, ±R) to appropriate accuracy 2 Create boxes containing zero lines Lemma (BOWW 2013) If C R 2 is convex, vol(c) < 1/2, then dim(c Z 2 ) 1. Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

11 Homogeneous Minimization Theorem - Quasiconvex/Quasiconcave division Let f be a homogeneous translatable polynomial of degree d 2 with integer coefficients. In polynomial time we can find a polynomial number of rational polyhedra P i, Q j and rational lines L k such that f is quasiconvex on P i, quasiconcave on Q j, and P Z 2 = l 1 i=1 P i l 2 j=1 Q j l 3 L k k=1 Z 2. (3) Theorem - Homogeneous, Bounded 1 In polynomial time, we can minimize a homogeneous translatable polynomial of fixed degree in 2 variables over the integer points of a bounded polyhedron. Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

12 Feasibility - Convex Set Operator Let C be a convex semi-algebraic set. In fixed dimension, we can determine in polynomial time if the following sets are non-empty: 1 (P C) Z n [KP, 2010] 2 (P \ C) Z n Enumerate vertices of P I [CKHM, 1993] Definition A division description of S f ω on P is a list of polyhedra P i, Q j and lines L k that cover P Z 2 such that 1 P i S f ω is convex 2 Q j S f >ω is convex Theorem Suppose P is bounded and that for every ω Z we can compute a division description of S f ω in polynomial time. Then, we can solve min{f(x) : x P Z n } in polynomial time. Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

13 Division Description for Quadratic-Cubic Consider the polynomial Then f(x, y) = 0 on curves f(x, y) = f 0 (x) + f 1 (x)y + f 2 (x)y 2 + f 3 (x)y 3 y ± = f 1 ± f 2 1 4f 0f 2 2f 0 1 Compute vertical asymptotes 2 Compute intersections of y + and y 3 Compute inflection points (changes in concavity) of y + and y Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

14 Division Description for Quadratic-Cubic Consider the polynomial Then f(x, y) = 0 on curves f(x, y) = f 0 (x) + f 1 (x)y + f 2 (x)y 2 + f 3 (x)y 3 y ± = f 1 ± f 2 1 4f 0f 2 2f 0 1 Compute vertical asymptotes 2 Compute intersections of y + and y 3 Compute inflection points (changes in concavity) of y + and y 4 Numerical approximations Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

15 Separating y + and y on intervals Lemma Any line can intersect a cubic level set at most 3 times. NP-hard Quartic Cubic Polynomials (x 2 ay + c) 2 Concave/Concave Concave/Convex Convex/Concave (Convex/Convex).... y + y y + y y + y. Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

16 Rotation of Cubic-cubic to Quadratic-cubic = x 3 + y u 2 v 3uv 2 + v = x 3 + δx 2 y + y u u 2 v 2.95uv 2 + v Lemma {(x, y) P Z 2 : f(x, y) 1/2} = {(x, y) P Z 2 : f ɛ (x, y) 1/2} Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

17 Rotation of Cubic-cubic to Quadratic-cubic = x 3 + y u 2 v 3uv 2 + v = x 3 + δx 2 y + y u u 2 v 2.95uv 2 + v Lemma {(x, y) P Z 2 : f(x, y) 1/2} = {(x, y) P Z 2 : f ɛ (x, y) 1/2} Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

18 Cubic Minimization (Bounded) Theorem - Cubic Division Description Let f be a cubic polynomial in 2 variables with integer coefficients and let ω Z In polynomial time, we can determine a division description of S f ω on P. Theorem - Cubic, Bounded 2 In polynomial time, we can minimize a cubic polynomial in 2 variables over the integer points of a bounded polyhedron. Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

19 Unbounded cubic-homogeneous Suppose h is a cubic homogeneous polynomial. (i) h(x, y) = 3 i=1 (a ix + b i y), (ii) h(x, y) = (a 1 x+b 1 y) 2 (a 2 x+b 2 y), (iii) h(x, y) = (a 1 x + b 1 y) 3, (iv) h(x, y) = (a 1 x + b 1 y) q(x, y) (i) (iii) (ii) (iv) Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

20 Unbounded cubic-homogeneous Suppose h is a cubic homogeneous polynomial. (i) h(x, y) = 3 i=1 (a ix + b i y), (ii) h(x, y) = (a 1 x+b 1 y) 2 (a 2 x+b 2 y), (iii) h(x, y) = (a 1 x + b 1 y) 3, (iv) h(x, y) = (a 1 x + b 1 y) q(x, y). Theorem - Cubic, Homogeneous, Unbounded 3 In polynomial time, we can solve (i) (iii) min{f(x) : x P Z 2 } with P possibly unbounded (ii) (iv) Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

21 Recap/Comments Theorem (Del Pia, H., Weismantel, Zemmer) The problem min{f(x) : x P Z 2 } can be solved in polynomial time in the following cases 1 f is fixed degree, homogeneous, P is bounded, 2 f is cubic, P is bounded*, 3 f is cubic, homogeneous, P is unbounded. Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

22 Recap/Comments Theorem (Del Pia, H., Weismantel, Zemmer) The problem min{f(x) : x P Z 2 } can be solved in polynomial time in the following cases 1 f is fixed degree, homogeneous, P is bounded, 2 f is cubic, P is bounded*, 3 f is cubic, homogeneous, P is unbounded. *Complete Complexity classification in dimension 2 for P bounded Homogeneous Result complexity not-necessarily dependent on the degree Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

23 Open Questions P Unbounded for general degree 3 in 2 variables Quadratic integer minimization in fixed dimension Classify all polynomials in dimension 2 where the optimization problem is solvable in polynomial time. Conjecture: All polynomials of fixed degree, dimension 2, whose gradient vanishes on a finite union of affine spaces, can be minimized over the integer points in a bounded polyhedron in polynomial time. Robert Hildebrand (ETH) Polynomial IP MINLP June 4, / 17

24 Thank you for listening!