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center-of-gravity method

{{One source|date=November 2023}}

The center-of-gravity method is a theoretic algorithm for convex optimization. It can be seen as a generalization of the bisection method from one-dimensional functions to multi-dimensional functions.{{Cite web |last=Nemirovsky and Ben-Tal |date=2023 |title=Optimization III: Convex Optimization |url=http://www2.isye.gatech.edu/~nemirovs/OPTIIILN2023Spring.pdf}}{{Rp|location=Sec.8.2.2}} It is theoretically important as it attains the optimal convergence rate. However, it has little practical value as each step is very computationally expensive.

Input

Our goal is to solve a convex optimization problem of the form:

minimize f(x) s.t. x in G,
where f is a convex function, and G is a convex subset of a Euclidean space Rn.

We assume that we have a "subgradient oracle": a routine that can compute a subgradient of f at any given point (if f is differentiable, then the only subgradient is the gradient \nabla f; but we do not assume that f is differentiable).

Method

The method is iterative. At each iteration t, we keep a convex region Gt, which surely contains the desired minimum. Initially we have G0 = G. Then, each iteration t proceeds as follows.

  • Let xt be the center of gravity of Gt.
  • Compute a subgradient at xt, denoted f'(xt).
  • By definition of a subgradient, the graph of f is above the subgradient, so for all x in Gt: f(x)−f(xt) ≥ (x−xt)Tf'(xt).
  • If f'(xt)=0, then the above implies that xt is an exact minimum point, so we terminate and return xt.
  • Otherwise, let Gt+1 := {x in Gt: (x−xt)Tf'(xt) ≤ 0}.

Note that, by the above inequality, every minimum point of f must be in Gt+1.{{Rp|location=Sec.8.2.2}}

Convergence

It can be proved that

Volume(G_{t+1})\leq \left[1-\left(\frac{n}{n+1}\right)^n\right]\cdot Volume(G_t) .
Therefore,
f(x_t) - \min_G f \leq \left[1-\left(\frac{n}{n+1}\right)^n\right]^{t/n} [\max_G f - \min_G f] .
In other words, the method has linear convergence of the residual objective value, with convergence rate \left[1-\left(\frac{n}{n+1}\right)^n\right]^{1/n} \leq (1-1/e)^{1/n} . To get an ε-approximation to the objective value, the number of required steps is at most 2.13 n \ln(1/\epsilon) + 1 .{{Rp|location=Sec.8.2.2}}

Computational complexity

The main problem with the method is that, in each step, we have to compute the center-of-gravity of a polytope. All the methods known so far for this problem require a number of arithmetic operations that is exponential in the dimension n.{{Rp|location=Sec.8.2.2}} Therefore, the method is not useful in practice when there are 5 or more dimensions.

See also

The ellipsoid method can be seen as a tractable approximation to the center-of-gravity method.

Instead of maintaining the feasible polytope Gt, it maintains an ellipsoid that contains it. Computing the center-of-gravity of an ellipsoid is much easier than of a general polytope, and hence the ellipsoid method can usually be computed in polynomial time.

References

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Category:Convex optimization