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Dual Phase Evolution pp 69–109Cite as

Problem Solving and Evolutionary Computation

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Abstract

Optimization algorithms impose an implicit network structure on fitness landscapes. For a given algorithm A operating on a problem that has a fitness landscape F, connections between solutions are defined by the transitions allowed by A.

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Notes

  1. 1.

    If we have a maximization problem, we can transform it to a minimization problem, e.g., by optimizing the negative of the objective function. In addition, an equation can be represented by two inequalities that hold at the same time: a “less than or equal” and a “greater than or equal” inequality. A “greater than or equal” inequality can be transformed to a “less than or equal” inequality by multiplying both sides by (\(-1\)). Thus, any optimization problem can be represented in the general form given in P1.

  2. 2.

    When the decision variable domain is unspecified, it will always be assumed to belong to \(\mathbb {R}\).

  3. 3.

    In the definitions of optimality (which we will give shortly) and in other definitions to follow, we will make extensive use of the open ball concept. This though does not mean that these definitions rely on open balls. Actually, they are also valid when more general notions of neighborhood are used. It is merely for the sake of striking a balance between illustrative clarity and mathematical accuracy that we will stick to the concept of open ball.

  4. 4.

    Ref. [1] defines the \(O\) notation as follows: Suppose \(f\) and \(g\) are positive functions defined for positive integers, \(f, \,\, g : I^+ \rightarrow R^+\), then

    1. 1.

      \(f=O(g)\), if  \(\exists \) positive constants \(c\) and \(N\) such that \(f(n) \le cg(n)\),    \(\forall \) \(n \ge N\).

    2. 2.

      \(f=\Omega (g)\), if  \(\exists \) positive constant \(c\) and \(N\) such that \(f(n) \ge cg(n)\),    \(\forall \) \(n \ge N\).

    3. 3.

      \(f=\Theta (g)\), if  \(\exists \) positive constant \(c\), \(d\), and \(N\) such that \(dg(n) \le f(n) \le cg(n)\),    \(\forall \) \(n \ge N\).

  5. 5.

    In optimization theory, a solution is an assignment of a value to each variable in the problem. This value is drawn from the domain of the corresponding variable defining the decision space. Each solution has an image when evaluated with the objective function(s) in the objective space. The use of the term “solution” in optimization theory is thus slightly counterintuitive; it differs from how we talk about “solutions” in our everyday language.

  6. 6.

    The values a variable can take is called the “domain” of the variable.

  7. 7.

    Remember, an LP problem is defined over a continuous domain; therefore, if the region of feasible solutions exists, it will be a continuous domain containing an infinite number of solutions.

  8. 8.

    Note that a genetic “algorithm” is not an algorithm as defined at the beginning of Sect. 3.3.1 because there is no mathematical proof that guarantees that a GA finds the optimum.

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Authors and Affiliations

  1. Faculty of Information Technology, Monash, Clayton, Australia

    David G. Green

  2. Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education, Xidian University, Xi’an, People’s Republic of China

    Jing Liu

  3. School of Engineering and Information Technology, University of New South Wales, Canberra, ACT, Australia

    Hussein A. Abbass

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Green, D.G., Liu, J., Abbass, H.A. (2014). Problem Solving and Evolutionary Computation. In: Dual Phase Evolution. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8423-4_3

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