# Tagged Questions

Convex Optimization is a special case of mathematical optimization. It includes Linear Programming and least-squares.

0answers
23 views

### Equivalence between standard optimization problem and Langragian form

Given a problem: $$\min_x f(x)$$ subject to $$g(x) \le C$$ In general, when it is equivalent to the problem $$\min_x f(x) + \lambda g(x)$$ for certain $\lambda$? Here my equivalence means : the ...
0answers
30 views

1answer
27 views

### minimum of sum of strictly convex functions

Is the following statement true? If so, how can I find a proof? Suppose that $f_1$ and $f_2$ are strictly convex functions on a convex set $X \subseteq \mathbb{R}^n$. If $f_1$ and $f_2$ have minimum,...
0answers
17 views

### how to calculate infimum of Augmented Lagrangian?

should any body explain that how do we calculate these step? \begin{align*} L(x,y) &= f(x) + y^T(Ax-b)\\ g(y) &= \inf_x \, L(x,y) \\ &= \inf_x \, f(x) - \langle -A^Ty, x \rangle - \langle ...
1answer
66 views

### Why is this matrix symmetric?

There is an example in the Convex Optimization lecture notes, Boyd. He just said in the lecture that the matrix which is underlined in red color is symmetric! How can we claim that when there is no ...
1answer
96 views

### Difference between Gradient Descent method and Steepest Descent

What is the difference between Gradient Descent method and Steepest Descent methods? In this book, they have come under different sections: http://stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf ...
1answer
41 views

### Gradient descent with linear perturbation

Given a convex, differentiable function $f$ (from a Hilbert space to $\mathbb{R}$) with a minimum (say $x^*$), I know you can find $x^*$ using gradient descent. Suppose now that you apply gradient ...
1answer
448 views

### Stochastic gradient descent for convex optimization

What happens if a convex objective is optimized by stochastic gradient descent? Is a global solution achieved?
1answer
66 views

2answers
379 views

### Gradient descent vs ternary search

Consider a strictly convex function $f: [0; 1]^n \rightarrow \mathbb{R}$. The question is why people (especially experts in machine learning) use gradient descent in order to find a global minimum of ...
1answer
563 views

### Using gradient descent and Newton's method combined

I have this function $f(\mathrm{X})$ where $\mathrm{X=A+B+C}$ where $\mathrm{A}$ is a diagonal element with variable $a$ on its diagonal. $\mathrm{B}$ is another diagonal matrix with variable $b$ on ...
0answers
9 views

2answers
28 views

### Convexity of Certain Functions

Consider the set of functions: $$f_n(t) := t^n e^{(\frac{c}{t^n})},$$ where $c$ is a non-zero real constant. I know that for $n=1$ $f_1(t)$ is convex on $(0,\infty)$ and ...
2answers
41 views

0answers
54 views

2answers
53 views

### Optimization with box constraints - via nonlinear function

I have the following convex optimization problem over $\mathbb{R}^n$ with box constraints: \begin{align}\text{minimize }&\;f(x)\\ \text{subject to }&\;x \in [-1,1]^n\end{align} I can see ...
2answers
16 views

1answer
40 views

### Is there a way to measure how (non)convex a function is, maybe analogous to condition number?

Consider the functions $f(x) = \sin x$ and $g(x) = (x+1)^2 (x-1)^2$. We know that $f$ has an infinite number of local minimizers and is nonconvex on a non-compact subset of $R$. We know that $g$ has ...
0answers
11 views

### Proof of the existence of a rational finitely generated cone

Let $P$ be a rational polyhedron and $F$ be the inclusion-wise minimal face. Then we define: \$C_F= \left\{c\in \mathbb{R}^n : F \subseteq \left\{x \in P:c^Tx=\max\left\{ c^Ty:y \in P\right\}\right\}\...