Extremum convexity

This section presents (i) the definitions and properties of convex and concave functions, (ii) the definitions of continuity, semicontinuity and subgradients, (iii) the definitions and properties of differentiable convex and concave functions, and (iv) the definitions and properties of local and global extremum points. 

Methods that use analytic-derivative information clearly possess more information about the smooth objective function. Gradient methods can use the slope of a function, for example, as the direction of movement toward extremum points. Second derivative methods can also incorporate curvature information from the Hessian to find the regions where the function is convex. 

A stationary point for a general Lagrangian function may or may not be a loctil extremum. If, as described in Section 2, suitable convexity conditions hold, then the method of Lagrange multipliers will yield a global minimum. 

Stoyan, Y.G., Novozhilova, M.V and Kartashov, A.V, 1996. Mathematical model and method of searching for a local extremum for the non-convex oriented polygons allocation problem. European Journal of Operational Research, 92,193-210. 

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