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Optimality Conditions for Unconstrained optimizationOne dimensional optimizationNecessary and sufficient conditionsMultidimensional optimizationClassification of stationary pointsNecessary and sufficient conditions for local optima.Convexity and global optimality

The topic of this lectures are necessary and sufficient conditions for a point to be an optimum. We will start with a review of the results known from Calculus for a function of single variable, in particular dealing with stationary points where the derivative is zero.

For one dimensional function, there are only three kinds of stationary points: maxima, minima, and inflection points. For multidimensional functions there are other kinds, so we will devote some time to their classification and to the necessary and sufficient conditions for local optima.

Finally, we will touch on conditions for global optimality, in particular for the special case of convex functions.1One dimensional optimizationWe are accustomed to think that if f(x) has a minimum then f(x)=0 but.

From Calculus we are accustomed to associate a minimum or a maximum with the vanishing of the first derivative. However, that applies only to differentiable functions with continuous derivatives as is illustrated in the figure.21D Optimization jargonA point with zero derivative is a stationary point. x=5, Can be a minimum

A maximum

An inflection point

For one dimensional functions stationary points are points where the derivative vanishes. We consider three functions that have a stationary point at x=5. One has a minimum ( Y=(x-5)2, the green curve), one has a maximum (y=10x-x2, the red curve), and one has an inflection point ( y=0.2(x-5)3, the blue curve).3Optimality criteria for smooth 1D functions at point x*f(x*)=0 is the condition for stationarity and a necessary condition for a minimum or a maximum.f(x*)>0 is sufficient for a minimumf(x*)0 is a sufficient condition for a minimum, because it guarantees that close enough to x* (h sufficiently small), f(x)>f*(x). Similarly, f(x*)=0 is called semi-positive definite, and all its eigenvalues are non-negative.7Types of stationary pointsPositive definite: MinimumPositive semi-definite: possibly minimumIndefinite: Saddle pointNegative semi-definite: possibly maximumNegative definite: maximum

So we can now offer a complete classification of stationary points for n-dimensional functions depending on the definiteness of the Hessian matrix. If the matrix is positive definite we have sufficient conditions for a minimum, and similarly if the matrix is negative definite (vTHv=0 for a minimum. In that case the matrix is called positive simi-definite, all the eigenvalues are non-negative, and if one ore more is zero, higher derivatives will determine whether we have a minimum.

Similarly, the necessary condition for a maximum, is that the matrix is negative semi-definite, which leads to the eigenvalues being non-positive.

Finally, if the matrix has vTHv>0 for some vectors v, and vTHv