A Refined Error Analysis for Fixed-Degree Polynomial Optimization over the Simplex

Zhao Sun 0 Mathematics Subject Classification 0 0 Z. Sun (&) Tilburg University , PO Box 90153, 5000 LE Tilburg, the Netherlands We consider the problem of minimizing a fixed-degree polynomial over the standard simplex. This problem is well known to be NP-hard, since it contains the maximum stable set problem in combinatorial optimization as a special case. In this paper, we revisit a known upper bound obtained by taking the minimum value on a regular grid, and a known lower bound based on Po lya's representation theorem. More precisely, we consider the difference between these two bounds and we provide upper bounds for this difference in terms of the range of function values. Our results refine the known upper bounds in the quadratic and cubic cases, and they asymptotically refine the known upper bound in the general case. - Global optimization 1 Introduction and Preliminaries Dn : fx 2 Rn : That is the global optimization problem f : mx2iDnn f x; or f : mx2aDxn f x: Here we focus on the problem of computing the minimum f of f over Dn. This problem is well known to be NP-hard, as it contains the maximum stable set problem as a special case (when f is quadratic). Indeed, given a graph G V ; E with adjacency matrix A, Motzkin and Straus [8] show that the maximum stability number aG can be obtained by e.g., [1, 5, 7] for fDn;r and [5, 14, 15] for fmrin d. The two ranges fDn;r f fmrin d have been studied separately and upper bounds for each of them have been shown in the above-mentioned works. In this paper, we study these two ranges at the same time. More precisely, we analyze the larger range fDn;r fmrin d and provide upper bounds for it in terms of the range of function values f f . Of course, upper bounds for the range fDn;r ranges fDn;r f and f fmrin d. Our new upper bound for fDn;r fmrin d refines these known bounds in the quadratic and cubic cases and provides an asymptotic refinement for general degree d. 1.1 Notation polynomials in n variables with degree d. For a 2 Nn, we denote xa : Qin1 xiai , while for I n , we let xI : Qi2I xi. Moreover, we denote xd : xx 1x 2 x d 1 for integer d > 0 and xa : Qin1 xiai for a 2 Nn. Thus, xd 0 if x is an integer with 0 6 x 6 d 1. for an integer r > 0. We define Dn; r : fx 2 Dn : rx 2 Nng; fDn;r : x2Dn;r min f x: Obviously, f 6 fDn;r 6 f and fDn;r can be computed by jDn; rj nrr 1 evaluations of f. In fact, when considering polynomials f of fixed degree d, the parameters fDn;r (with increasing values of r) provide a PTAS for (1. 1), as was proved by Bomze and de Klerk [1] (for d 2), and by de Klerk et al. [5] (for d > 2). Recently, de Klerk et al. [7] provide an alternative proof for this PTAS and refine the error bound for fDn;r f from [5] for cubic f. In addition, some researchers study the properties of the regular grid Dn; r. For instance, given a point x 2 Dn, Bomze et al. [2] show a scheme to find the closest point to x on Dn; r with respect to some class of norms including p-norms for p > 1. 1.3 Lower Bounds Based on Polyas Representation Theorem has nonnegative coefficients. Notice that f can be equivalently formulated as f max k s:t: f x Then, one can easily check the following inequalities: 0 1 fm in 6 fm in 6 Lemma 1.1 For f Pb2In;d fbxb 2 Hn;d, one has 10 1 d! xcA@ X fbxbA Hence, by Definition (1.2), we obtain k > 0 8a 2 In; r 1.4 Bernstein Coefficients For any polynomial f Pb2In;d fbxb 2 Hn;d, we can write it as For any b 2 In; d, we call fb bd!! the Bernstein coefficients of f (this terminology has also been used in [4, 7]), since they are the coefficients of the polynomial f when f is expressed in the Bernstein basis fbd!! xb : b 2 In; dg of Hn;d. Applying the multinomial theorem together with (1.4), one can obtain that when evaluating f at a point x 2 Dn, f x is a convex combination of the Bernstein coefficients fb bd!!. Therefore, we have Theorem 1.2 [5, Theorem 2.2] For any polynomial f P one has 1.5 Contribution of the Paper bounds for fDn;r [7]) and for f f , and by adding them up one can easily derive range mink6r fDn;k fmrin d in terms of f f . Our results in this paper refine the results in [5, 7, 15] for the quadratic and cubic cases (see Sects. 2 and 3 respectively), while for the general case, our result refines the result of [5] when r is sufficiently large (see Sect. 5). 2 The Quadratic Case For any quadratic polynomial f, we consider the range fDn;r following upper bound in terms of f f . Theorem 2.1 For any quadratic f xTQx and r > 2, one has where Qmax : maxi2n Qii. Proof By (1.3), we have Hence, r r 1 fmrin 2 mina2In;r f ar 1 Xn Now we point out that our result (2.1) refines the relevant result of [5]. De Klerk et al. [5] show the following theorem. Theorem 2.2 [5, Theorem 3.2] Suppose f 2 Hn;2 and r > 2. Then 1.6 Structure The paper is organized as follows. In Sects. 2 and 3, we consider the quadratic and cubic cases, respectively and refine the relevant results obtained from [5, 7, 15]. Then, we look at the square-free (aka multilinear) case in Sect. 4. Moreover, in Sect. 5, we consider general (fixed-degree) polynomials and compare our new result with the one of [5]. By adding up (2.3) and (2.4), one gets which is implied by our result (2.1). Moreover, in [15], Yildirim considers one hierarchical upper bound of f (when f is quadratic), which is defined by mink6r fDn;k: One can easily verify that fmrin 2 6 f 6 min fDn;k 6 fDn;r: k6r In [15, Theorem 4.1], Yildirim shows mink6r fDn;k f , which can also be easily implied by our result (2.1). The following example shows that the upper bound (2.1) can be tight. Example 2.3 [7, Example 2] Consider the quadratic polynomial f Pin1 xi2. As f is convex, one can check that f n1 (attained at x n1 e) and f 1 (attained at any standard unit vector). To compute fDn;r, we write r as r kn s, where k > 0 and 0 6 s\n. Then one can check that By (1.3), we have 3 The Cubic Case Theorem 3.1 Proof For any cubic polynomial f, we consider the difference fDn;r following result. We can write any cubic polynomial f as f Xn fixi3 X i1 i\j fijxixj2 gijxi2xj fijkxixjxk: Then by (1.3), one can check that r 1r2r 2fmrin 3 a2mIinn;r a f r 3 Xfiai2 3 Xf 1 ai 2 > fDn;r r a2mIanx;r i r i1 1 min 2Xn f ai r2 a2In;r i1 i r 1 max(3 Xnfixi2 X > fDn;r r x2Dn i1 i\j 2 Xnfiai X fij gijaiaj 1 n min2 Xfixi: fij gijxixj r2 x2Dn i1 fi fj fij gij 6 8f : Using (3.4) and the fact that Pn i1 xi 1, one can obtain fij gijxixj 6 By (3.2), (3.3), (3.5) and the fact that Pn i1 xi 1, one can get Hence, one has 2fDn;r r 2f 6 4rf Now we observe that our result (3.1) refines the relevant upper bound obtained from [5, 7]. De Klerk et al. [5] show the following result. Theorem 3.2 [5, Theorem 3.3] Suppose f 2 Hn;3 and r > 3. Then Recently, De Klerk et al. [7, Corollary 2 ] refine (3.7) to Similar to the quadratic case (in Sect. 2), our new upper bound (3.1) implies the upper bound obtained by adding up (3.6) and (3.8). (...truncated)