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Luus–Jaakola
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<p>In <a href="/facts/Computational_engineering/dvIJ5o36">computational engineering</a>, Luus–Jaakola (LJ) denotes a <a href="/facts/Heuristic_algorithm/PM2qkoa2">heuristic</a> for <a href="/facts/Global_optimization/nksWlDqU">global</a> <a href="/facts/Optimization_(mathematics)/oRn8Iv5I">optimization</a> of a real-valued function. In engineering use, LJ is not an <a href="/facts/Algorithm/fnl5NmRt">algorithm</a> that terminates with an optimal solution; nor is it an <a href="/facts/Iterative_method/uKMAJhEh">iterative method</a> 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, <a href="/facts/Newton%27s_method/VlRhI2FL">Newton's method</a> 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 <a href="/facts/Convex_function/IbzG5SLF">convex</a> nor <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a> nor <a href="/facts/Lipschitz_continuity/Hw20EPEU">locally Lipschitz</a>: The LJ heuristic does not use a <a href="/facts/Gradient/TsR7uukj">gradient</a> or <a href="/facts/Subgradient/IK3s3Cqw">subgradient</a> when one be available, which allows its application to non-differentiable and non-convex problems. </p><p>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 <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">uniform distribution</a>. With each iteration, the neighborhood decreases, which forces a subsequence of iterates to converge to a cluster point. </p><p>Luus has applied LJ in <a href="/facts/Optimal_control/9SmcQHjJ">optimal control</a>, <a href="/facts/Transformer/EYYfXWrF">transformer design</a>, <a href="/facts/Metallurgy/1SANyxX4">metallurgical processes</a>, and <a href="/facts/Chemical_engineering/Fb64amQf">chemical engineering</a>. </p>