Intereting Posts

Derivative of double integral with respect to upper limits
What is the limit of this divergent infinite product multiplied by an exponential?
Multiple Fourier Integrals involving Heaviside Theta Function
Who named “Quotient groups”?
Why can't you take the limit of a 2-D function in every direction and call that the limit if they're equal?
Split up sum of products $\sum{a_i b_i}\approx(1/N)\sum{a_i}\sum{b_i}$ for uncorrelated summands?
Find splitting field of a cubic polynomial
About the determination of complex logarithm
Minimum area of Inscribed Square
Does a function have to be “continuous” at a point to be “defined” at the point?
Examples when vector $(X,Y)$ is not normal 2D distribution, but X and Y are.
Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$?
Number of cycles in complete graph
Can a nice enough ODE always be extended to the complex plane?
Big O notation sum rule

Let’s say I would like to minimize a convex function $f(x)$ over a set $C$. $C$ is not convex but a union of a finite number of convex sets $C_i$: $C = C_1 \cup \dots \cup C_m$ where each $C_i$ is convex. Then can I minimize $f(x)$ over each $C_i$ and take the minimum of the results to obtain the minimum of $f(x)$ over $C$?

Also, is it true that any bounded set $C \subset \mathbb{R}^n$ can be written as a union of a finite number of convex sets?

- Subadditivity of the $n$th root of the volume of $r$-neighborhoods of a set
- Dual norm intuition
- Projection of a point onto a convex polyhedra
- Subtracting a constant from log-concave function preserves log-concavity, if the difference is positive
- Epigraph of closed convex hull of a function
- Two fundamental questions about convexity of a function (number1)

- Summation notation problem
- Show that the maximum of a set of convex functions is again convex
- The distribution of barycentric coordinates
- What is the optimal path between $2$ fixed points around an invisible obstructing wall?
- Connected-ness of the boundary of convex sets in $\mathbb R^n$ , $n>1$ , under additional assumptions of the convex set being compact or bounded
- Why does gradient ascent/descent have zig-zag motion?
- When does it help to write a function as $f(x) = \sup_\alpha \phi_{\alpha}(x)$ (an upper envelope)?
- Maximum of product of numbers when the sum is fixed
- Is the convex hull of closed set in $R^{n}$ is closed?
- How to solve the matrix minimization for BFGS update in Quasi-Newton optimization

The answer to the first question is basically yes. But you must be careful – you say “minimum” for example, rather than “infimum,” but the minimum of a convex function over a convex set is not always attained. Consider, for example, minimizing $e^x$ over $(0,1)$. On the other hand, it is true that $$\inf_{C} ~f(x) = \min_{i=1, \ldots, m} ~~\inf_{C_i} ~~f(x).$$

The answer to the second question is no. Let $C$ be the set of rational numbers in $[0,1]$; then $C$ cannot be written as a union of finitely many convex sets. Indeed, a convex set in $R$ which is not a singleton is necessarily an interval; if your union of finitely many convex sets includes at least one interval, you’ve included some irrational numbers in there and the result can’t equal $C$. The other case – when your union does not have any intervals, i.e., its all singletons – will not work either because $C$ has infinitely many elements while a finite union of singletons only has finitely many elements.

- Prove that $\arctan\left(\frac{2x}{1-x^2}\right)=2\arctan{x}$ for all $|x|<1$, directly from the integral definition of $\arctan$
- Are all connected manifolds homogeneous
- Does there exist a non-empty set that is a subset of its power set?
- Tricks to remember Vector Calculus formulas
- Methods for choosing $u$ and $dv$ when integrating by parts?
- Integral of Bessel function multiplied with sine
- What is the minimum polynomial of $x = \sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6} = \cot (7.5^\circ)$?
- Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$
- Will $2$ linear equations with $2$ unknowns always have a solution?
- Transient diffusion with compact support throughout (not just initially)
- Explicit formula for inverse matrix elements
- Simple probability problems
- Is “monotonous” ever used as a synonym for “monotonic” in math?
- Finding the closure of some subsets of the ordered square
- Order of the centralizer of a permutation