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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]
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.
Characteristic function (convex analysis) Closed convex function; Complex convexity; Concave function; Concavification; Convex cap; Convex compactification; Convex cone; Convex conjugate; Convex function; Convex hull; Convex optimization; Convex set; Copositive matrix
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]
Cardinal function; Cauchy-continuous function; Closed convex function; Coarse function; Coercive function; Comparison function; Concave function; Constructible function; Continuous function; Continuous functions on a compact Hausdorff space; Convex function; Cyclical monotonicity
Otherwise, if a feasible solution exists and if the constraint set is bounded, then the optimum value is always attained on the boundary of the constraint set, by the maximum principle for convex functions (alternatively, by the minimum principle for concave functions) since linear functions are both convex and concave. However, some problems ...
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.