Local unimodal sampling Source: en.wikipedia.org/wiki/Local_unimodal_sampling

In computational engineering, Luus–Jaakola (LJ) denotes a heuristic for global optimization of a real-valued function.^[1] In engineering use, LJ is not an algorithm that terminates with an optimal solution; nor is it an iterative method that generates a sequence of points that converges to an optimal solution (when one exists). However, when applied to a twice continuously differentiable function, the LJ heuristic is a proper iterative method, that generates a sequence that has a convergent subsequence; for this class of problems, Newton's method is recommended and enjoys a quadratic rate of convergence, while no convergence rate analysis has been given for the LJ heuristic.^[1] In practice, the LJ heuristic has been recommended for functions that need be neither convex nor differentiable nor locally Lipschitz: The LJ heuristic does not use a gradient or subgradient when one be available, which allows its application to non-differentiable and non-convex problems.

Proposed by Luus and Jaakola,^[2] LJ generates a sequence of iterates. The next iterate is selected from a sample from a neighborhood of the current position using a uniform distribution. With each iteration, the neighborhood decreases, which forces a subsequence of iterates to converge to a cluster point.^[1]

Luus has applied LJ in optimal control,^{[3] [4]} transformer design,^[5] metallurgical processes,^[6] and chemical engineering.^[7]

At each step, the LJ heuristic maintains a box from which it samples points randomly, using a uniform distribution on the box. For a unimodal function, the probability of reducing the objective function decreases as the box approach a minimum. The picture displays a one-dimensional example.

Let ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ be the fitness or cost function which must be minimized. Let ${\displaystyle {\textbf {x}}\in \mathbb {R} ^{n}}$ designate a position or candidate solution in the search-space. The LJ heuristic iterates the following steps:

• Initialize x ~ U(b_lo,b_up) with a random uniform position in the search-space, where b_lo and b_up are the lower and upper boundaries, respectively.
• Set the initial sampling range to cover the entire search-space (or a part of it): d = b_up − b_lo
• Until a termination criterion is met (e.g. number of iterations performed, or adequate fitness reached), repeat the following:
  • Pick a random vector a ~ U(−d, d)
  • Add this to the current position x to create the new potential position y = x + a
  • If (f(y) < f(x)) then move to the new position by setting x = y, otherwise decrease the sampling-range: d = 0.95 d
• Now x holds the best-found position.

Luus notes that ARS (Adaptive Random Search) algorithms proposed to date differ in regard to many aspects.^[8]

• Procedure of generating random trial points.
• Number of internal loops (NIL, the number of random search points in each cycle).
• Number of cycles (NEL, number of external loops).
• Contraction coefficient of the search region size. (Some example values are 0.95 to 0.60.)
  • Whether the region reduction rate is the same for all variables or a different rate for each variable (called the M-LJ algorithm).
  • Whether the region reduction rate is a constant or follows another distribution (e.g. Gaussian).
• Whether to incorporate a line search.
• Whether to consider constraints of the random points as acceptance criteria, or to incorporate a quadratic penalty.

Nair proved a convergence analysis. For twice continuously differentiable functions, the LJ heuristic generates a sequence of iterates having a convergent subsequence.^[1] For this class of problems, Newton's method is the usual optimization method, and it has quadratic convergence (regardless of the dimension of the space, which can be a Banach space, according to Kantorovich's analysis).

The worst-case complexity of minimization on the class of unimodal functions grows exponentially in the dimension of the problem, according to the analysis of Yudin and Nemirovsky, however. The Yudin-Nemirovsky analysis implies that no method can be fast on high-dimensional problems that lack convexity:

"The catastrophic growth [in the number of iterations needed to reach an approximate solution of a given accuracy] as [the number of dimensions increases to infinity] shows that it is meaningless to pose the question of constructing universal methods of solving ... problems of any appreciable dimensionality 'generally'. It is interesting to note that the same [conclusion] holds for ... problems generated by uni-extremal [that is, unimodal] (but not convex) functions."^[9]

When applied to twice continuously differentiable problems, the LJ heuristic's rate of convergence decreases as the number of dimensions increases.^[10]

• Random optimization is a related family of optimization methods that sample from general distributions, for example the uniform distribution.
• Random search is a related family of optimization methods that sample from general distributions, for example, a uniform distribution on the unit sphere.
• Pattern search are used on noisy observations, especially in response surface methodology in chemical engineering. They do not require users to program gradients or hessians.