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A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
For example, the problem of maximizing a concave function can be re-formulated equivalently as the problem of minimizing the convex function . The problem of maximizing a concave function over a convex set is commonly called a convex optimization problem. [8]
This follows from the fact that the convolution of two log-concave functions is log-concave. The product of two log-concave functions is log-concave. This means that joint densities formed by multiplying two probability densities (e.g. the normal-gamma distribution, which always has a shape parameter ≥ 1) will be log-concave.
In simple terms, a convex function graph is shaped like a cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap . A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. [1]
Fractional programming studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem. Nonlinear programming studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex ...
A quasiconvex function that is not convex A function that is not quasiconvex: the set of points in the domain of the function for which the function values are below the dashed red line is the union of the two red intervals, which is not a convex set. The probability density function of the normal distribution is quasiconcave but not concave.
In mathematics, a sequence a = (a 0, a 1, ..., a n) of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if a i 2 ≥ a i−1 a i+1 holds for 0 < i < n. Remark: some authors (explicitly or not) add two further conditions in the definition of log-concave sequences: a is non-negative
We start the conjugate gradient iteration by computing the residual := = =, of the Schur complement system, where := denotes the upper half of the solution vector matching the initial guess for its lower half.