Some additional properties of elementary landscapes (original) (raw)

Applied Mathematics Letters, 2003

When joined to a stipulated neighborhood digraph, an objective function defined on the solution space of a real combinatorial optimization problem forms a landscape. Grover shows that landscapes satisfying a certain difference equation have properties favorable to local search.

Quasi-elementary landscapes and superpositions of elementary landscapes

2012

There exists local search landscapes where the evaluation function is an eigenfunction of the graph Laplacian that corresponds to the neighborhood structure of the search space. Problems that display this structure are called "Elementary Landscapes" and they have a number of special mathematical properties. The term "Quasi-elementary landscapes" is introduced to describe landscapes that are "almost" elementary; in quasi-elementary landscapes there exists some efficiently computed "correction" that captures those parts of the neighborhood structure that deviate from the normal structure found in elementary landscapes. The "shift" operator, as well as the "3-opt" operator for the Traveling Salesman Problem landscapes induce quasi-elementary landscapes. A local search neighborhood for the Maximal Clique problem is also quasi-elementary. Finally, we show that landscapes which are a superposition of 2 elementary landscapes are also quasi-elementary in structure.

Tunably Rugged Landscapes with Known Maximum and Minimum

IEEE Transactions on Evolutionary Computation, 2015

We propose NM landscapes as a new class of tunably rugged benchmark problems. NM landscapes are well-defined on alphabets of any arity, including both discrete and real-valued alphabets, include epistasis in a natural and transparent manner, are proven to have known value and location of the global maximum and, with some additional constraints, are proven to also have a known global minimum. Empirical studies are used to illustrate that, when coefficients are selected from a recommended distribution, the ruggedness of NM landscapes is smoothly tunable and correlates with several measures of search difficulty. We discuss why these properties make NM landscapes preferable to both NK landscapes and Walsh polynomials as benchmark landscape models with tunable epistasis.

The component model for elementary landscapes and partial neighborhoods

Theoretical Computer Science, 2014

Local search algorithms exploit moves on an adjacency graph of the search space. An "elementary landscape" exists if the objective function f is an eigenfunction of the Laplacian of the graph induced by the neighborhood operator; this allows various statistics about the neighborhood to be computed in closed form. A new component based model makes it relatively simple to prove that certain types of landscapes are elementary. The traveling salesperson problem, weighted graph (vertex) coloring and the minimum graph bisection problem yield elementary landscapes under commonly used local search operators. The component model is then used to efficiently compute the mean objective function value over partial neighborhoods for these same problems. For a traveling salesperson problem over n cities, the 2-opt neighborhood can be decomposed into n/2 − 1 partial neighborhoods. For graph coloring and the minimum graph bisection problem, partial neighborhoods can be used to focus search on those moves that are capable of producing a solution with a strictly improving objective function value.

Barrier Trees of Degenerate Landscapes

Zeitschrift für Physikalische Chemie, 2000

The heights of energy barriers separating two (macro-)states are useful for estimating transition frequencies. In non-degenerate landscapes the decomposition of a landscape into basins surrounding local minima connected by saddle points is straightforward and yields a useful definition of macro-states. In this work we develop a rigorous concept of barrier trees for degenerate landscapes. We present a program that efficiently computes such barrier trees, and apply it to two well known examples of landscapes.

Extrema Constrained by a Family of Curves and Local Extrema

Journal of Optimization Theory and Applications - J OPTIMIZ THEOR APPL, 1998

This paper considers the connections between the local extrema of a function f:D?R and the local extrema of the restrictions of f to specific subsets of D. In particular, such subsets may be parametrized curves, integral manifolds of a Pfaff system, Pfaff inequations. The paper shows the existence of C1 or C2-curves containing a given sequence of points. Such curves are then exploited to establish the connections between the local extrema of f and the local extrema of f constrained by the family of C1 or C2-curves. Surprisingly, what is true for C1-curves fails to be true in part for C2-curves. Sufficient conditions are given for a point to be a global minimum point of a convex function with respect to a family of curves.

Scaling, Optimality, and Landscape Evolution

Journal of Statistical …, 2001

A nonlinear model is studied which describes the evolution of a landscape under the effects of erosion and regeneration by geologic uplift by mean of a simple differential equation. The equation, already in wide use among geomorphologists and in that context obtained phenomenologically is here derived by reparametrization invariance arguments and exactly solved in dimension d=1. Results of numerical simulations in d=2 show that the model is able to reproduce the critical scaling characterizing landscapes associated with natural river basins. We show that configurations minimizing the rate of energy dissipation (optimal channel networks) are stationary solutions of the equation describing the landscape evolution. Numerical simulations show that a careful annealing of the equation in the presence of additive noise leads to configurations very close to the global minimum of the dissipated energy, characterized by mean field exponents. We further show that if one considers generalized river network configurations in which splitting of the flow (i.e., braiding) and loops are allowed, the minimization of the dissipated energy results in spanning loopless configurations, under the constraints imposed by the continuity equations. This is stated in the form of a general theorem applicable to generic networks, suggesting that other branching structures occurring in nature may possibly arise as optimal structures minimizing a cost function.

A direct approach to Plateau's problem in any codimension

Advances in Mathematics, 2016

This paper proposes a direct approach to solve the Plateau's problem in codimension higher than one. The problem is formulated as the minimization of the Hausdorff measure among a family of d-rectifiable closed subsets of R n : following the previous work [DGM14], the existence result is obtained by a compactness principle valid under fairly general assumptions on the class of competitors. Such class is then specified to give meaning to boundary conditions. We also show that the obtained minimizers are regular up to a set of dimension less than (d -1).

A note on König's minimax theorem

Acta Mathematica Hungarica, 1994

Recently G. Kassay [1] published an elementary proof of Khnig's minimax theorem [2]. His method seems to be an interesting mixture of both of the so-called methods of level sets and cones, respectively. Formally, Khnig's theorem is an extension of Ky Fan's classical minimax theorem [3] by restricting convexity to diadic rational convexity. It is well-known [4] that Ky Fan's theorem can be deduced from the Br6zis-Nirenberg-Stampacchia level set minimax theorem by a function lifting. It is an old open question whether there is a short direct connection between Khnig's and Ky Fan's minimax theorems. The aim of this note is to show that the mentioned function lifting in [4] transforms a Khnig-type saddle function into a Ky Fan-type saddle function with the same minimax values. A careful analysis of the proof of this fact leads also to new generalizations of Khnig's theorem, which seem not be provable with a simple adaptation of Kassay's method. Finally we remark that the question of K6nig-type generalizations of M. Sion's minimax theorem [5] is still open.

Epistasis as a Basic Concept in Formal Landscape Analysis

Icga, 1997

In this note we measure the interdependency of bits in the encoding of a fitness function using two different invariants. The first invariant, normalized epistasis, features a strong correlation with GA-hardness, as we illustrate here with generalized Royal Road functions. The second one, bit decidability, imitates epistasis, but has the avantage of being less difficult to calculate and approximate. Although it seems less powerful than epistasis, it is equally able to rank fitness functions according to their GA-hardness. We prove that the function class E 3 , which is similar to the class of NK( ; 2) landscapes, is NPequivalent. This class is based on a generalization of the concept of epistasis, and, because of the latter theorem, its structure is believed to be important in landscape analysis.