Because x and y are variables that are both x and y in a utility function, an indifference curve is convex to the origin if the derivative is always negative and the second derivative is positive.
What does it mean when a function is convex?
If the line segment connecting any two points on the graph of a real-valued function does not lie below the graph between the two points, the function is said to be convex in mathematics. A function is convex if its epigraph (the set of points on or above the function’s graph) is also a convex set. A single-variable twice-differentiable function is convex if and only if its second derivative is nonnegative across its entire domain. The quadratic function is a well-known example of a single-variable convex function.
What is the best way to show that a utility function is quasiconcave?
If f(x) is greater than f(x’), then f((1x + x’) is more than f(x’). A function is rigorously quasiconcave if every point on any line segment connecting points on two level curves produces a higher value for the function than any point on the level curve corresponding to the lower value of the function, except the endpoints.
How do you determine a function’s concavity and convexity?
Let’s start by determining whether the graph is increasing or decreasing. We’ll need the first derivative for that.
We can apply the power rule to find the first derivative. We multiply by the original variable and lower the exponent on all the variables by one.
Fill in the blanks with the value we’ve been given. The function is growing if the outcome is positive. The function is decreasing if the outcome is negative.
Look at the second derivative to discover the concavity. The function is concave if it is positive at our given point. It is convex if the function is negative.
We repeat the method, but this time utilizing as our phrase to determine the second derivative.
Our second derivative is a constant, as you can see. Our output will always be negative, regardless of which point we plug in for. As a result, our graph will be convex at all times.
When we combine our two pieces of knowledge, we can observe that the graph is decreasing and convex at the given position.
Why does the utility function have a concave shape?
Concavity can be found in both ordinal and cardinal utility functions. Concavity is a feature of utility functions that appears to be independent of ordinal or cardinal assumptions. It is derived from the notion that preferences are convex.
Convex Optimization Problems
A convex optimization problem is one in which all of the constraints are convex functions, and the objective is either a convex or a concave function depending on whether you’re minimizing or maximizing. Linear programming issues are convex because linear functions are convex. Convex issues are the natural extension of conic optimization problems, which are the natural extension of linear programming problems. In a convex optimization problem, the feasible zone is a convex region, which is defined as the intersection of convex constraint functions, as shown below.
There can only be one optimal solution, which is globally optimal, with a convex objective and a convex feasible region.
Several methods, including Interior Point methods, will either find the globally optimal solution or prove that the problem is unsolvable.
Up to a very big size, convex problems can be addressed efficiently.
Any issue in which the objective or any of the constraints are non-convex, as shown below, is referred to as a non-convex optimization problem.
Multiple feasible regions and multiple locally optimal points within each region may exist in such a situation.
Determining that a non-convex issue is infeasible, that the objective function is unbounded, or that an optimal solution is the “global optimum” across all feasible regions can require time exponential in the number of variables and constraints.
Convex Functions
A function is convex geometrically if a line segment drawn from any point (x, f(x)) to another point (y, f(y)) known as the chord from x to y lies on or above the graph of f, as shown below:
In algebra, f is convex if f(tx + (1-t)y) is concave if -f is convex that is, if the chord from x to y sits on or below the graph of f for any x and y and any t between 0 and 1.
Every linear function, whose graph is a straight line, is both convex and concave, as can be seen.
It is neither convex nor concave, hence a non-convex function “curves up and down.”
The sine function is a well-known example:
However, from -pi to 0, this function is convex, and from 0 to +pi, it is concave.
The overall issue is convex if the variable bounds restrict the scope of the objective and constraints to an area where the functions are convex.
Solving Convex Optimization Problems
Convex optimization issues can be solved using a variety of strategies due to their favorable qualities. Interior Point or Barrier methods, on the other hand, are particularly well suited to convex problems because they treat linear, quadratic, conic, and smooth nonlinear functions in essentially the same way: they create and use a smooth convex nonlinear barrier function for the constraints, even for LP problems.
These approaches make it possible to solve convex problems of any size up to a very high size, and they are especially useful for second-order (quadratic and SOCP) problems with constant Hessians. Interior Point techniques, regardless of the number of variables and restrictions, require a relatively small number of iterations (usually less than 50) to obtain an optimal solution, according to both theoretical and empirical results (though the computational effort per iteration rises with the number of variables and constraints). Hardware advancements, such as instruction caching, pipelining, and other modifications in processor architecture, have favored Interior Point techniques more than other approaches.