# Tagged Questions

37 views

### How to show $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$ for $\theta \in [0,1]$?

Let $\theta \in [0, 1]$. Let $f(x,y)$ be a function. Is there a way I could prove that $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$? I have tried to start with $f(x,y) = 2f(x,y) - f(x,y)$ or ...
28 views

31 views

261 views

### Maximization of sum of two functions

Is there any relationship similar to the following. Let $X$ be the maximum of functions $f_1(x)+f_2(x)$. Let $X_1$ be a maximum of $f_1(x)$ and let $X_2$ be a maximum of $f_2(x)$. Is there any ...
107 views

### Constrained maximization problem

I need help with the following optimization problem $$\max\;\alpha\ln(x(1-y^2))+(1-\alpha)\ln(z)$$ where the maximization is with respect to $x,y,z$, subject to \begin{align} \alpha ...
191 views

### How to find $\kappa$ to minimize integral $I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) \mathrm{d}x$

I am trying to find such value $\kappa \in (0,1)$ that would minimize the integral \begin{aligned} I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) ...
267 views

### Global Min-Max Optimization

When is $$\min_X \max_Y f(X,Y)$$ globally solvable? (i.e. we can find global solution for the optimization problem?) I am not looking for reformulations. Is it only when ...
211 views

Take two real differentiable convex functions, $f_1$ and $f_2$, defined on the unit interval $[0; 1]$. I want to find the global optimum of: $\min_{x \in [0;1]} af_1(x)+bf_2(x)$, for given $a, b \in ... 2answers 178 views ### Simple example where strong convexity is necessary over strict (or even regular) convexity. I am trying to get a hold on exactly what strong convexity gives you over strict (or regular) convexity. Yes there are simple functions which demonstrate the difference between these ideas, but what ... 2answers 171 views ### Entropy expression optimization with Langrange multipliers I have recently encountered variants of the following expression: $$S = H(a,b,c,d)-H(a+b,c+d)$$ where$H$is the Shannon entropy function, that is$H(X)=\sum_{x\in X}-x\log ...
Consider the following function, $$f(x, y) = e^{m e^{-y}+n e^{-x}-x-y} \left(a x e^y+b e^x y+c x y\right)$$ where $a, b, c, m$ and $n$ are positive constants. I want to show $f(x, y)$ is ...