Fixed-point computation


Fixed-point computation refers to the process of computing an exact or approximate fixed point of a given function.[1] In its most common form, we are given a function f that satisfies the condition to the Brouwer fixed-point theorem, that is: f is continuous and maps the unit d-cube to itself. The Brouwer fixed-point theorem guarantees that f has a fixed point, but the proof is not constructive. Various algorithms have been devised for computing an approximate fixed point. Such algorithms are used in economics for computing a market equilibrium, in game theory for computing a Nash equilibrium, and in dynamic system analysis.

Definitions Edit

The graph of an example function with three fixed points

The unit interval is denoted by  , and the unit d-dimensional cube is denoted by Ed. A continuous function f is defined on Ed (from Ed to itself). Often, it is assumed that f is not only continuous but also Lipschitz continuous, that is, for some constant L,   for all x,y in Ed.

A fixed point of f is a point x in Ed such that f(x) = x. By the Brouwer fixed-point theorem, any continuous function from Ed to itself has a fixed point. But for general functions, it is impossible to compute a fixed point precisely, since it can be an arbitrary real number. Fixed-point computation algorithms look for approximate fixed points. There are several criteria for an approximate fixed point. Several common criteria are:[2]

  • The residual criterion: given an approximation parameter   , An ε-residual fixed-point of f is a point x in Ed such that  , where here |.| denotes the maximum norm. That is, all d coordinates of the difference   should be at most ε.[3]: 4 
  • The absolute criterion: given an approximation parameter  , A δ-absolute fixed-point of f is a point x in Ed such that  , where   is any fixed-point of f.
  • The relative criterion: given an approximation parameter  , A δ-relative fixed-point of f is a point x in Ed such that  , where   is any fixed-point of f.

For Lipschitz-continuous functions, the absolute criterion is stronger than the residual criterion: If f is Lipschitz-continuous with constant L, then   implies  . Since   is a fixed-point of f, this implies  , so  . Therefore, a δ-absolute fixed-point is also an ε-residual fixed-point with  .

The most basic step of a fixed-point computation algorithm is a value query: given any x in Ed, the[sentence fragment]

The function f is accessible via evaluation queries: for any x, the algorithm can evaluate f(x). The run-time complexity of an algorithm is usually given by the number of required evaluations.

Contractive functions Edit

A Lipschitz-continuous function with constant L is called contractive if L < 1; it is called weakly-contractive if L ≤ 1.Every contractive function satisfying Brouwer's condisions has a unique fixed point. Moreover, fixed-point computation for contractive functions is easier than for general functions.

Computing a fixed point using function iteration

The first algorithm for fixed-point computation was the fixed-point iteration algorithm of Banach. Banach's fixed-point theorem implies that, when fixed-point iteration is applied to a contraction mapping, the error after t iterations is in  . Therefore, the number of evaluations required for a δ-relative fixed-point is approximately  . Sikorski and Wozniakowski[4] showed that Banach's algorithm is optimal when the dimension is large. Specifically, when  , the number of required evaluations of any algorithm for δ-relative fixed-point is larger than 50% the number of evaluations required by the iteration algorithm. Note that when L approaches 1, the number of evaluations approaches infinity. In fact, no finite algorithm can compute a δ-absolute fixed point for all functions with L=1.[5]

When L < 1 and d = 1, the optimal algorithm is the Fixed Point Envelope (FPE) algorithm of Sikorski and Wozniakowski.[4] It finds a δ-relative fixed point using   queries, and a δ-absolute fixed point using   queries. This is much faster than the fixed-point iteration algorithm.[6]

When d>1 but not too large, and L ≤ 1, the optimal algorithm is the interior-ellipsoid algorithm (based on the ellipsoid method).[7] It finds an ε-residual fixed-point is using   evaluations. When L<1, it finds a δ-absolute fixed point using   evaluations.

Shellman and Sikorski[8] presented an algorithm called BEFix (Bisection Envelope Fixed-point) for computing an ε-residual fixed-point of a two-dimensional function with L ≤ 1, using only   queries. They later[9] presented an improvement called BEDFix (Bisection Envelope Deep-cut Fixed-point), with the same worst-case guarantee but better empirical performance. When L<1, BEDFix can also compute a δ-absolute fixed-point using   queries.

Shellman and Sikorski[2] presented an algorithm called PFix for computing an ε-residual fixed-point of a d-dimensional function with L ≤ 1, using   queries. When L < 1, PFix can be executed with  , and in that case, it computes a δ-absolute fixed-point, using   queries. It is more efficient than the iteration algorithm when L is close to 1. The algorithm is recursive: it handles a d-dimensional function by recursive calls on (d-1)-dimensional functions.

Algorithms for differentiable functions Edit

When the function f is differentiable, and the algorithm can evaluate its derivative (not only f itself), the Newton method can be used and it is much faster.[10][11]

General functions Edit

For functions with Lipschitz constant L > 1, computing a fixed-point is much harder.

One dimension Edit

For a 1-dimensional function (d = 1), a δ-absolute fixed-point can be found using   queries using the bisection method: start with the interval  ; at each iteration, let x be the center of the current interval, and compute f(x); if f(x) > x then recurse on the sub-interval to the right of x; otherwise, recurse on the interval to the left of x. Note that the current interval always contains a fixed point, so after   queries, any point in the remaining interval is a δ-absolute fixed-point of f. Setting  , where L is the Lipschitz constant, gives an ε-residual fixed-point, using   queries.[3]

Two or more dimensions Edit

For functions in two or more dimensions, the problem is much more challenging. Shellman and Sikorski[2] proved that, for any integers d ≥ 2 and L > 1, finding a δ-absolute fixed-point of d-dimensional L-Lipschitz functions might require infinitely-many evaluations. The proof idea is as follows. For any integer T > 1, and for any sequence of T of evaluation queries (possibly adaptive), one can construct two functions that are Lipschitz-continuous with constant L, and yield the same answer to all these queries, but one of them has a unique fixed-point at (x, 0) and the other has a unique fixed-point at (x, 1). Any algorithm using T evaluations cannot differentiate between these functions, so cannot find a δ-absolute fixed-point. This is true for any finite integer T.

Several algorithms based on function evaluations have been developed for finding an ε-residual fixed-point

  • The first algorithm to approximate a fixed point of a general function was developed by Herbert Scarf in 1967.[12][13] Scarf's algorithm finds an ε-residual fixed-point by finding a fully-labelled "primitive set", in a construction similar to Sperner's lemma.
  • A later algorithm by Harold Kuhn[14] used simplices and simplicial partitions instead of primitive sets.
  • Developing the simplicial approach further, Orin Harrison Merrill[15] presented the restart algorithm.
  • B. Curtis Eaves[16] presented the homotopy algorithm. The algorithm works by starting with an affine function that approximates f, and deforming it towards f, while following the fixed point. A book by Michael Todd[1] surveys various algorithms developed until 1976.
  • David Gale[17] showed that computing a fixed point of an n-dimensional function (on the unit d-dimensional cube) is equivalent to deciding who is the winner in an d-dimensional game of Hex (a game with d players, each of whom needs to connect two opposite faces of an d-cube). Given the desired accuracy ε
    • Construct a Hex board of size kd, where  . Each vertex z corresponds to a point z/k in the unit n-cube.
    • Compute the difference f(z/k) - z/k; note that the difference is an n-vector.
    • Label the vertex z by a label in 1, ..., d, denoting the largest coordinate in the difference vector.
    • The resulting labeling corresponds to a possible play of the d-dimensional Hex game among d players. This game must have a winner, and Gale presents an algorithm for constructing the winning path.
    • In the winning path, there must be a point in which fi(z/k) - z/k is positive, and an adjacent point in which fi(z/k) - z/k is negative. This means that there is a fixed point of f between these two points.

In the worst case, the number of function evaluations required by all these algorithms is exponential in the binary representation of the accuracy, that is, in  .

Query complexity Edit

Hirsch, Papadimitriou and Vavasis proved that[3] any algorithm based on function evaluations, that finds an ε-residual fixed-point of f, requires   function evaluations, where   is the Lipschitz constant of the function   (note that  ). More precisely:

  • For a 2-dimensional function (d=2), they prove a tight bound  .
  • For any d ≥ 3, finding an ε-residual fixed-point of a d-dimensional function requires   queries and   queries.

The latter result leaves a gap in the exponent. Chen and Deng[18] closed the gap. They proved that, for any d ≥ 2 and   and  , the number of queries required for computing an ε-residual fixed-point is in  .

Discrete fixed-point computation Edit

A discrete function is a function defined on a subset of   (the d-dimensional integer grid). There are several discrete fixed-point theorems, stating conditions under which a discrete function has a fixed point. For example, the Iimura-Murota-Tamura theorem states that (in particular) if f is a function from a rectangle subset of   to itself, and f is hypercubic direction-preserving, then f has a fixed point.

Let f be a direction-preserving function from the integer cube   to itself. Chen and Deng[18] prove that, for any d ≥ 2 and n > 48d, computing such a fixed point requires   function evaluations.

Chen and Deng[19] define a different discrete-fixed-point problem, which they call 2D-BROUWER. It considers a discrete function f on   such that, for every x on the grid, f(x) - x is either (0, 1) or (1, 0) or (-1, -1). The goal is to find a square in the grid, in which all three labels occur. The function f must map the square  to itself, so it must map the lines x = 0 and y = 0 to either (0, 1) or (1, 0); the line x = n to either (-1, -1) or (0, 1); and the line y = n to either (-1, -1) or (1,0). The problem can be reduced to 2D-SPERNER (computing a fully-labeled triangle in a triangulation satisfying the conditions to Sperner's lemma), and therefore it is PPAD-complete. This implies that computing an approximate fixed-point is PPAD-complete even for very simple functions.

Relation between fixed-point computation and root-finding algorithms Edit

Given a function g from Ed to R, a root of g is a point x in Ed such that g(x)=0. An ε-root of g is a point x in Ed such that  .

Fixed-point computation is a special case of root-finding: given a function f on Ed, define  . Clearly, x is a fixed-point of f if and only if x is a root of g, and x is an ε-residual fixed-point of f if and only if x is an ε-root of g. Therefore, any root-finding algorithm (an algorithm that computes an approximate root of a function) can be used to find an approximate fixed-point.

The opposite is not true: finding an approximate root of a general function may be harder than finding an approximate fixed point. In particular, Sikorski[20] proved that finding an ε-root requires   function evaluations. This gives an exponential lower bound even for a one-dimensional function (in contrast, an ε-residual fixed-point of a one-dimensional function can be found using   queries using the bisection method). Here is a proof sketch.[3]: 35  Construct a function g that is slightly larger than ε everywhere in Ed except in some small cube around some point x0, where x0 is the unique root of g. If g is Lipschitz continuous with constant L, then the cube around x0 can have a side-length of  . Any algorithm that finds an ε-root of g must check a set of cubes that covers the entire Ed; the number of such cubes is at least  .

However, there are classes of functions for which finding an approximate root is equivalent to finding an approximate fixed point. One example[18] is the class of functions g such that   maps Ed to itself (that is:   is in Ed for all x in Ed). This is because, for every such function, the function   satisfies the conditions to Brouwer's fixed-point theorem. Clearly, x is a fixed-point of f if and only if x is a root of g, and x is an ε-residual fixed-point of f if and only if x is an ε-root of g. Chen and Deng[18] show that the discrete variants of these problems are computationally equivalent: both problems require   function evaluations.

Communication complexity Edit

Roughgarden and Weinstein[21] studied the communication complexity of computing an approximate fixed-point. In their model, there are two agents: one of them knows a function f and the other knows a function g. Both functions are Lipschitz continuous and satisfy Brouwer's conditions. The goal is to compute an approximate fixed point of the composite function  . They show that the deterministic communication complexity is in  .

References Edit

  1. ^ a b The Computation of Fixed Points and Applications. Lecture Notes in Economics and Mathematical Systems. Vol. 124. 1976. doi:10.1007/978-3-642-50327-6. ISBN 978-3-540-07685-8.[page needed]
  2. ^ a b c Shellman, Spencer; Sikorski, K. (December 2003). "A recursive algorithm for the infinity-norm fixed point problem". Journal of Complexity. 19 (6): 799–834. doi:10.1016/j.jco.2003.06.001.
  3. ^ a b c d Hirsch, Michael D; Papadimitriou, Christos H; Vavasis, Stephen A (December 1989). "Exponential lower bounds for finding Brouwer fix points". Journal of Complexity. 5 (4): 379–416. doi:10.1016/0885-064X(89)90017-4. S2CID 1727254.
  4. ^ a b Sikorski, K; Woźniakowski, H (December 1987). "Complexity of fixed points, I". Journal of Complexity. 3 (4): 388–405. doi:10.1016/0885-064X(87)90008-2.
  5. ^ Sikorski, Krzysztof A. (2001). Optimal Solution of Nonlinear Equations. Oxford University Press. ISBN 978-0-19-510690-9.[page needed]
  6. ^ Sikorski, K. (1989). "Fast Algorithms for the Computation of Fixed Points". Robustness in Identification and Control. pp. 49–58. doi:10.1007/978-1-4615-9552-6_4. ISBN 978-1-4615-9554-0.
  7. ^ Huang, Z; Khachiyan, L; Sikorski, K (June 1999). "Approximating Fixed Points of Weakly Contracting Mappings". Journal of Complexity. 15 (2): 200–213. doi:10.1006/jcom.1999.0504.
  8. ^ Shellman, Spencer; Sikorski, K. (June 2002). "A Two-Dimensional Bisection Envelope Algorithm for Fixed Points". Journal of Complexity. 18 (2): 641–659. doi:10.1006/jcom.2001.0625.
  9. ^ Shellman, Spencer; Sikorski, K. (September 2003). "Algorithm 825: A deep-cut bisection envelope algorithm for fixed points". ACM Transactions on Mathematical Software. 29 (3): 309–325. doi:10.1145/838250.838255. S2CID 7786886.
  10. ^ Kellogg, R. B.; Li, T. Y.; Yorke, J. (September 1976). "A Constructive Proof of the Brouwer Fixed-Point Theorem and Computational Results". SIAM Journal on Numerical Analysis. 13 (4): 473–483. doi:10.1137/0713041.
  11. ^ Smale, Steve (July 1976). "A convergent process of price adjustment and global newton methods". Journal of Mathematical Economics. 3 (2): 107–120. doi:10.1016/0304-4068(76)90019-7.
  12. ^ Scarf, Herbert (September 1967). "The Approximation of Fixed Points of a Continuous Mapping". SIAM Journal on Applied Mathematics. 15 (5): 1328–1343. doi:10.1137/0115116.
  13. ^ H. Scarf found the first algorithmic proof: Voitsekhovskii, M.I. (2001) [1994], "Brouwer theorem", Encyclopedia of Mathematics, EMS Press, ISBN 1-4020-0609-8.
  14. ^ Kuhn, Harold W. (1968). "Simplicial Approximation of Fixed Points". Proceedings of the National Academy of Sciences of the United States of America. 61 (4): 1238–1242. doi:10.1073/pnas.61.4.1238. JSTOR 58762. PMC 225246. PMID 16591723.
  15. ^ Merrill, Orin Harrison (1972). Applications and Extensions of an Algorithm that Computes Fixed Points of Certain Upper Semi-continuous Point to Set Mappings (Thesis). OCLC 570461463. NAID 10006142329.
  16. ^ Eaves, B. Curtis (December 1972). "Homotopies for computation of fixed points". Mathematical Programming. 3–3 (1): 1–22. doi:10.1007/BF01584975. S2CID 39504380.
  17. ^ Gale, David (1979). "The Game of Hex and Brouwer Fixed-Point Theorem". The American Mathematical Monthly. 86 (10): 818–827. doi:10.2307/2320146. JSTOR 2320146.
  18. ^ a b c d Chen, Xi; Deng, Xiaotie (2005). "On algorithms for discrete and approximate brouwer fixed points". Proceedings of the thirty-seventh annual ACM symposium on Theory of computing. pp. 323–330. doi:10.1145/1060590.1060638. ISBN 1581139608. S2CID 16942881.
  19. ^ Chen, Xi; Deng, Xiaotie (October 2009). "On the complexity of 2D discrete fixed point problem". Theoretical Computer Science. 410 (44): 4448–4456. doi:10.1016/j.tcs.2009.07.052. S2CID 2831759.
  20. ^ Sikorski, K. (June 1984). "Optimal solution of nonlinear equations satisfying a Lipschitz condition". Numerische Mathematik. 43 (2): 225–240. doi:10.1007/BF01390124. S2CID 120937024.
  21. ^ Roughgarden, Tim; Weinstein, Omri (2016). "On the Communication Complexity of Approximate Fixed Points". 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS). pp. 229–238. doi:10.1109/FOCS.2016.32. ISBN 978-1-5090-3933-3. S2CID 87553.

Further reading Edit

  • Yannakakis, Mihalis (May 2009). "Equilibria, fixed points, and complexity classes". Computer Science Review. 3 (2): 71–85. doi:10.1016/j.cosrev.2009.03.004.