Convex Optimization 2026, 24 - Constraints & Fitting Problems

Given a point v in D, the interpolation conditions f(v_j) = z_j form a set of linear equalities. The Lipschitz constraint, for example forms a set of linear inequalities in x. Constraints can be applied to an infinite number of points, as long as it's a subset of D. Such constraint may be convex, but generally don't lead to tractable problems (p. 329). If the basis functions are differentiable at a point v ∈ D, the gradient of f(v) is a linear function of x, and interpolation conditions on the derivative reduce to linear equality constraints. Lipschitz-boundedness of its gradient is a convex constraint. Similarly, constraints on the gradient can just as well be imposed on a set of infinite points (p. 330). Any linear functional on the subspace of functions can be expressed as a linear function. This includes integrals (p. 330 - 331, 6.5.2).

The data for a fitting problem (u, y) that seeks a function that matches it. In such cases, least-square fitting could be applied, for example. In that case, the problem would minimize ∑(f(u_i - y_i))². Simple function fitting problems will have many more data points than the dimension of the subspace of functions. All members of that subspace are smooth (p. 333). Function fitting with fewer data points than dimension of subspace of functions, can, in the simplest case, require the function satisfy interpolation conditions. It may filter for smoothest or smallest target function. These ultimately lead to least-norm problems (p. 333). By evaluating the optimal function fit at a point v outside the original set, one may start with an initial guess, and optimized in fitting steps.