1. If you know calculus, take the second derivative. It is a well-known fact that if the second derivative f (x) is ≥ 0 for all x in an interval I, then f is convex on I. On the other hand, if f(x) ≤ 0 for all x ∈ I, then f is concave on I.
How do you prove a function is convex optimization?
Geometrically, a function is convex if a line segment drawn from any point (x, f(x)) to another point (y, f(y)) -- called the chord from x to y -- lies on or above the graph of f, as in the picture below: Algebraically, f is convex if, for any x and y, and any t between 0 and 1, f( tx + (1-t)y ) \x26lt;= t f(x) + (1-t) f(y)..
What is a convex value function?
A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval..
What is the optimal value function?
The optimal Value function is one which yields maximum value compared to all other value function. When we say we are solving an MDP it actually means we are finding the Optimal Value Function..
A function f : Rn → R is convex if and only if the function g : R → R given by g(t) = f(x + ty) is convex (as a univariate function) for all x in domain of f and all y ∈ Rn. (The domain of g here is all t for which x + ty is in the domain of f.) Proof: This is straightforward from the definition.
Feb 4, 2016U ⊂ Rn is nonempty, closed and convex with 0 ∈ U (not nec. poly.) B ∈ Rn×n is symmetric positive semi-definite. Examples: 1. Support
Abstract. Convexity and concavity properties of the optimal value functionf* are considered for the general parametric optimization problemP(ɛ) of the form min
The optimal value function of a convex model generally is not continuous and it is not known analytically. Still, in some situations, it is possible to obtain enough information about it in order to calculate and describe its local and global optima.
Are convex functions a problem?
One has for convex functions the same technical problem as for convex sets: all convex functions that occur in applications have the nice properties of being closed and proper, but if you work with them and make new functions out of them, then they might lose these properties
In a convex optimization problem, x ∈ Rn is a vector known as the optimization variable, f : Rn → R is a convex function that we want to minimize, and C ⊆ Rn is a convex set describing the set of feasible solutions.
Microeconomic function
In microeconomics, a consumer's Marshallian demand function is the quantity they demand of a particular good as a function of its price, their income, and the prices of other goods, a more technical exposition of the standard demand function. It is a solution to the utility maximization problem of how the consumer can maximize their utility for given income and prices. A synonymous term is uncompensated demand function, because when the price rises the consumer is not compensated with higher nominal income for the fall in their real income, unlike in the Hicksian demand function. Thus the change in quantity demanded is a combination of a substitution effect and a wealth effect. Although Marshallian demand is in the context of partial equilibrium theory, it is sometimes called Walrasian demand as used in general equilibrium theory.
Convex optimal value function
Concept in game theory
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a unique distribution of a total surplus generated by the coalition of all players. The Shapley value is characterized by a collection of desirable properties. Hart (1989) provides a survey of the subject.
