Feasible Region Approximation Using Convex Polytop es Sachin S: Sapatnekar Pravin M: Vaidya S: M: Kang Department of EE/CprE Department of Computer Science Co ordinated Science Lab oratory Iowa State University University of Illinois University of Illinois Ames, IA 50011. Urbana, IL 61801. Urbana, IL 61801. Abstract A p olytop e can b e thought of as the convex hull of its vertices. Hence, it is capable of representing any convex A new technique for p olytop e approximation of the b o dy in the limiting case where the numberofvertices feasible region for a design is presented. This metho d and hence, hyp erplanes is in nite. A satisfactory ap- is computationally less exp ensive than the simplicial proximation to the convex b o dy can b e obtained using a approximation metho d [1]. Results on several circuits nite number of hyp erplanes. are presented, and it is shown that the qualityofthe In this work, we address the problem of feasible region p olytop e approximation is substantially b etter than approximation by convex p olytop es. We present a new an ellipsoidal approximation. approach that is computationally cheap er than the exist- I. Introduction ing approachin[1], and illustrate how p olytop e approxi- mation is sup erior to ellipsoidal approximation. While designing a circuit according to certain sp eci - cations, it is useful to de ne a feasible region as the set of II. Approximation of the Feasible Region parameter values for which the design sp eci cations are satis ed. It is a commonly-made assumption [1, 2] that A. Introduction the feasible region is a convex b o dy; while in general, this n The feasible region FR , where n is the number of is not true, it is usually true that the feasible region is design parameters, is formally de ned as the set of p oints \nearly convex." in the design parameter space for which the circuit satis- An example of a situation in which the concept of the es all sp eci cations on its b ehavior. It is often assumed feasible region is useful is the problem of design center- that F is a convex body. In this work, we preserve that ing. Realising that pro cess variations may cause param- assumption in de ning the approachtobetaken, and use eter values to drift from their nominal values, this prob- prop erties of convex sets to create an approximation to lem seeks to select the nominal values of design param- F . In our implementation though, we showhowwemay eters in suchaway that the b ehavior of the circuit re- make allowances for the incorrectness of this assumption. mains within sp eci cations with the greatest probability. The simplicial approximation metho d [1] provides a In other words, the aim of design centering is to ensure metho d for approximating a feasible region by a p oly- that the manufacturing yield is maximized. An approach top e. Each iteration of this metho d involves the solution that is commonly used to solve this problem [1, 2] approx- of f + 1 linear programs where f is the numb er of faces of imates the feasible region by a convex body, and takes the the p olytop e, a line search, and up dating a convex hull. design center to b e its exact or estimated center. Apart from the fact that each linear program solution is The pro cedure in this pap er can b e used for the approx- exp ensive, the numb er of linear programs to b e solved in imation of any convex or \nearly convex" b o dy. the entire pro cedure is large sp eci cally, k f + 1, where Twotyp es of convex b o dies have b een used most com- k is the numb er of iterations. Moreover, the pro cedure monly for feasible region approximation: of up dating the convex hull is also computationally ex- i An el lipsoid, which is a b o dy de ned by p ensive. This is b ecause up dating a convex hull consists T 2 nn n of identifying hyp erplanes to b e removed and nding the fy j y y B y y r g; B2R ; y ; y 2 R : 1 c c c equations of the new hyp erplanes. Obtaining the equation Ellipsoids are inherently limited in approximating asym- 3 of a hyp erplane, given the vertices, is an On op eration; metric regions, and regions with linear edges. this needs to b e carried out for each new hyp erplane of the ii A polytope, which is de ned as an intersection of half- p olytop e while up dating the convex hull in an iteration. spaces, is given by In the algorithm presented here, an initial p olytop e is mn m P = fx j Ax bg;A 2 R ; b 2 R : 2 up dated by p erforming a set of line searches. After up 1 to 2n line searches, a new p olytop e center has to b e com- k=0 2:5 Initialize P and z puted, whichisanOn op eration. Thus, the algorithm 0 0 While planes added in the last iteration 6= 0 { that we present is computationally less complex than sim- P = P plicial approximation. k +1 k Cho ose orthogonal directions d ; d d B. Construction of the Approximating Polytope 1 2 n For i =1ton { Our algorithm is based on the following well-known the- Perform a binary search along direction d i orem on convex sets [3] that is stated b elow. from z to nd a b oundary p oint z of F 0 b;i Supp orting Hyp erplane Theorem : Let C b e a con- If z exists, and z 2P , and b;i b;i k +1 vex set and let y b e a b oundary p ointofC . Then there distancez ,b oundary of P > { b;i k +1 exists a hyp erplane containing y and containing C in one Addahyp erplane to the p olytop e of its closed half-spaces. P = P [ new hyp erplane k +1 k +1 De nition : A tangent plane at a p oint z on the b ound- 0 } ary of a region is given by Perform a binary search along direction d i T T from z to nd a b oundary p oint z of F 0 b;i [rg z ] z =[rg z ] z 3 0 0 0 If z exists, and z 2P , and b;i b;i k +1 distancez ,b oundary of P > { b;i k +1 where gz 0 is an active constraint at p oint z , and 0 Addahyp erplane to the p olytop e rg . is the gradient of the function g .. P = P [ new hyp erplane k +1 k +1 For a p oint z on the b oundary of a convex set, C ,a 0 } tangent plane at z satis es the supp orting hyp erplane 0 } prop erty. In particular, for the constraint g z 0, C is Set z = center of the up dated p olytop e, P 0 k +1 contained in the half-space k = k +1; T T } [rg z ] z [rg z ] z : 4 0 0 0 The pro cedure requires a technique for calculating the The aim of the algorithm is to approximate the feasible gradient of an active constraint at a b oundary p oint. As n region, FR by a p olytop e in [2], one of several metho ds may b e employed for this purp ose. If the exact functional form of the constraintis mn m P = fz j Az bg;A 2 R ; b 2 R ; 5 known, an expression for the gradientmay b e derived. If not, metho ds such as the adjoint network technique [5]or n formed by the intersection of m half-spaces in R . nite di erences may b e used to calculate the gradient. The algorithm b egins with an initial feasible p oint, n n III. Experimental Results z 2F R .Ann-dimensional b ox, fz 2 R j z 0 min z z g, containing F is chosen as the initial p oly- i max A. A Numerical Example top e P . In each iteration, n orthogonal search directions, 0 Anumerical example in two dimensions is presented d ; d d are chosen. A binary search is conducted 1 2 n here to illustrate our technique. The feasible region is from z to identify a b oundary p oint z of F , for each 0 b;i represented by the ellipse direction d .Ifz is relatively deep in the interior of P , i b;i 2 1 T then the tangent plane to F at z is added to the set b;i z j z t z t 1 7 F = 1 2 of constraining hyp erplanes in Equation 5. In practice, since the feasible region is not strictly convex, it is useful T with the ellipse center, t,at[4; 4] . The approximating to shift the tangent plane awayby a distance factor .In p olytop e, P , shown in Fig. 1, can b e seen to b e a go o d other words, if c is the gradient of the active constraint approximation of the ellipse. at the b oundary p oint, the hyp erplane added is B.ATunable Active Filter The techniques describ ed in Section I I were used to ap- T T c z c z 1 6 b;i proximate the feasible region for a tunable active lter shown in Fig.