page_title: Luus–Jaakola
section_title: Motivation
hierarchical_section_title: Luus–Jaakola / Motivation
caption_attribution_description: English: In optimization of a single-dimensional unimodal function by random sampling with a uniform distribution, when the current position x is far from the optimum the probability is 1/2 for finding an improvement.
context_page_description: In computational engineering, Luus–Jaakola denotes a heuristic for global optimization of a real-valued function. 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. 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. 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, 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.
context_section_description: 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.
caption: When the current position x is far from the optimum the probability is 1/2 for finding an improvement through uniform random sampling.
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