Center-of-gravity method

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.^{[1]: 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 ${\displaystyle \nabla f}$; but we do not assume that f is differentiable).

Method

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

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

Note that, by the above inequality, every minimum point of f must be in G_t+1.^{[1]: Sec.8.2.2}

Convergence

It can be proved that

${\displaystyle Volume(G_{t+1})\leq \left[1-\left({\frac {n}{n+1}}\right)^{n}\right]\cdot Volume(G_{t})}$ .

Therefore,

${\displaystyle f(x_{t})-\min _{G}f\leq \left[1-\left({\frac {n}{n+1}}\right)^{n}\right]^{t/n}[\max _{G}f-\min _{G}f]}$.

In other words, the method has linear convergence of the residual objective value, with convergence rate ${\displaystyle \left[1-\left({\frac {n}{n+1}}\right)^{n}\right]^{1/n}\leq (1-1/e)^{1/n}}$. To get an ε-approximation to the objective value, the number of required steps is at most ${\displaystyle 2.13n\ln(1/\epsilon )+1}$.^{[1]: 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.^{[1]: 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 G_t, 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