All Questions
Tagged with nonlinear-programming or nonlinear-optimization
639 questions
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95
views
How to maximum L1 norm problem?
I have met a problem these days.
\begin{equation}
\underset{\omega}{\max} \quad \Vert \text{diag}(\mathbf{h}^H)\mathbf{G}^H\mathbf{\omega}\Vert_1 \\
s.t.\quad\mathbf{\omega}^H\mathbf{G}\mathbf{G}^H\...
- 171
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votes
0
answers
44
views
Is there a multiplier rule for this minimization problem?
Let $(E,\mathcal E)$ be a measurable space, $W\subseteq\left\{w:E\to\mathbb R\mid w\text{ is }\mathcal E\text{-measurable}\right\}$ be a Banach space, $k\in\mathbb N$ and $f:W^k\to[0,\infty)$. I'm ...
- 167
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0
answers
32
views
Spectral measures of a family of parameter-dependent self-adjoint contractions on an $L^2$-space
I have a self-adjoint linear contraction $A_g$ on an $L^2$-space of the form $$A_gf=\int\gamma(f,g),$$ where $\gamma$ is Lipschitz continuous and $g$ is an a priori fixed function. Assuming $1-A_g$ is ...
- 167
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0
answers
96
views
About martingales induced by iterative processes
Suppose I have a discrete stochastic process $\{ X_i \}_{i=1,\ldots..}$ defined as, $X_{i+1} = X_i - \eta \nabla f(X_i) + \sqrt{\eta} \xi_i$ where $f : \mathbb{R}^d \rightarrow \mathbb{R}$ and $\xi_i \...
- 2,246
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votes
0
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121
views
Closed form solution to an equation
Let $X \in \mathbb{R}^{n \times d}, w \in \mathbb{R}^d, y \in \{\pm 1
\}^{n}, \alpha \in (0, 1)$. Consider the equation
$$ X^{\top}(Xw-y)+\alpha \|w\|_{2}X^{\top}\operatorname{sign}(Xw-y)+\alpha\frac{...
0
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0
answers
101
views
How can we analytically solve this max-sum-min problem?
Let $I$ be a finite set, and $A_{ij},B_{ij},x_i,y_j\ge0$. I want to find the choice of $x_i,y_j$ maximizing $$\sum_{i\in I}\sum_{j\in J}A_{ij}\min\left(x_i,B_{ij}y_j\right)\tag1$$ subject to $$\sum_{i\...
- 167
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votes
0
answers
65
views
How to find the implicit solution of PDE's?
I wanted to know if there were hints and educated guesses to find the implicit solutions of complicated PDE's.
I'm currently dealing with functions of the form $f(r,t)$.
I know that a lot of time a ...
- 121
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answers
45
views
Sensitivity of optimization solutions?
I've been working with the following optimization problem:
$$ \max \int \left(\frac{1}{2}\left\| x \right\|^2 + \tilde{u}(x)\right) \,ds(x) + \int \left(\frac{1}{2}\left\| y \right\|^2 + \tilde{v}(y) ...
- 383
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73
views
Number of local minima of a particular non-convex composite function
I am interested in proving that the following equation on the interval $x \in [c,1]$ is minimized either at the endpoints or where $x=\sqrt{c}$:
$f(x)=\frac{-1}{\log(1-x)}+\frac{-1}{\log(1-\frac{c}{x}...
0
votes
1
answer
246
views
Cross entropy loss is not twice differentiable?
I was reading a recent theory paper in machine learning by Kenji Kawaguchi and Leslie Pack Kaelbling
https://arxiv.org/pdf/1901.00279.pdf
and the authors seem to suggest in section 2.2 that cross-...
- 283
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votes
0
answers
181
views
non-convex optimization with constraint
I have a special non-convex optimization problem:
$\min / \max \ f(x) + g(x) + h(x)$,
subject to $| g(x) - h(x)| < \varepsilon$,
where $f(x)$ is non-convex, but both $g(x)$ and $h(x)$ are ...
- 43
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0
answers
66
views
Suggestions to solve an optimization problem that involves quadratic forms
I am in a crucial part of my research, I have arrived at an optimization problem that I can not solve, I need to solve it to be able to perform simulations and thus complete my research, due to this ...
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0
answers
84
views
Number of solutions to non-linear equations
As part of our project, we are required to determine the total number of distinct solutions to the following equations.
There are $n$ variables of one type, say $\{p_i\}_{i=1}^n$, $m$ variables of ...
- 109
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votes
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answers
198
views
Maximum entropy with constraint on CDF
I would like to know whether the following problem is well posed, and whether there is a solution. Let me make it clear that I have no pretentions of rigor here.
Given a continuum random variable $x$...
- 343
0
votes
2
answers
144
views
Optimization function of two variables
Let $A, B, C, D \in \mathbb{R^*_+}$.
Is it possible to solve
$$
\max_{ \substack{0 \leq x\leq A \\ 0\leq y\leq B}} \frac{1+x+y}{(1+Cx)(1+Dy)}
$$
The KKT conditions give for an extrema $(x^*,y^*)$
...
user83947
0
votes
0
answers
42
views
What (analytical or numerical) method can I use to solve scalar optimal problem?
I got the following optimization problem in mind and I am looking for some (analytic or numerical) methods to solve it. Can anyone have any ideas? Here is problem
\begin{aligned}
& {\text{...
- 221
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votes
0
answers
181
views
Mixed-Integer Bilinear Program (MIBLP)
Consider the problem of
\begin{align}
\min_{x,y} \quad &a^Tx + b^Ty + x^TQy \\
&Ax \leq d \\
&Cy \leq e \\
&x_i \in \mathbb{R} \quad i \in \{1,2,\ldots,N\} \\
&y_i \in {\{0,1\}} \...
- 113
0
votes
0
answers
180
views
Question on solving an optimization problem using Variable splitting and ADMM
Tell me if I have found the right approach to the following optimization problem:
$$
min_{x} \frac{1}{2}\left \| Ax-b \right \|_2^2
\\
s.t. \ \ \Phi v=x \ , \ {x^T(1-x)}=0
$$
$A$ and $\Phi$ ...
- 109
0
votes
0
answers
2k
views
Interior point V.S sqp algorithm for large scale optimization
I have an large scale optimization problem that works with fmincon solver within sqp algorithm. But it is so slow. However, <...
- 101
0
votes
0
answers
47
views
Complexity of optimizing a bi-objective function with integer constraints
I have two different objective functions, each of which can be solved optimally in polynomial time. Does this mean, I can optimize a linear combination of these objective functions in polynomial time ...
- 121
0
votes
0
answers
444
views
How to decide a value of learning rate for Stochastic Gradient Descent?
I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation,
$w_{i+1} = w_i + -\eta \nabla E_n(...
0
votes
0
answers
42
views
Optimization problem involving an entrywise function
Let $X$ a $n\times p$ real-valued matrix and $Y$ a $p\times q$ real-valued matrix. Let $\phi:\mathbb{R} \to \mathbb{R}$ a function. What is the appropriate way to deal with the following optimization ...
- 840
0
votes
0
answers
80
views
Finding gradient of an optimization
I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me?
Assume that we have an optimization ...
- 161
0
votes
0
answers
61
views
Solution of a nonlinear system of two equations
Given the matrix $A_{M,N}$ with $N\gt M$, the vector $y$, I have to find the vectors $x$ and $u$, satisfying the following equations:
$$D(x)x=A^Tu$$
$$y=Ax$$
where: $$D(x) = \left| \begin{array}{ccc}
\...
- 399
0
votes
0
answers
81
views
A Optimization problem using co-ordinates of joint numerical range.
Let $\mathbf{A}_1,\dots,\mathbf{A}_L$ be $N\times N$ hermitian matrices. Define the mapping from the $N-$dimensional unit sphere to $\mathbb{R}^L$ as
\begin{align}
\mathcal{S}=\{\left(\mathbf{u}^H\...
- 1,421
0
votes
0
answers
91
views
Complexity of turning a d-degree polynomial to 2-degree polynomial
For a very simple example,
$(1+x)^4=x^4+4x^3+6x^2+4x+1$ is a 4 degree polynomial, and I want to change it to a 2-degree polynomial by add more variables, for this example, we can simply let $y=x^2$, ...
- 187
0
votes
0
answers
533
views
Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$
Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function.
Also, the member of $X$ are uniformly bounded
$$
|x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}.
$$
Let $F: X \...
user24451
0
votes
0
answers
104
views
Big eigenvalues of a special stochastic matrix
Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq |\lambda_2|(M)....
- 105
0
votes
0
answers
189
views
functional maximization
Define a functional space of functions of the form $F(t)=p_1 exp^{-\mu_1(\delta-t)}+p'_1 (1-exp^{-\mu_1(\delta-t)}))$. $p_1,p'_1,\delta,\mu$ are parameters in [0,1] and trivially, variation of these ...
- 221
0
votes
0
answers
67
views
Minimizing inside a spherical uncertainty region
I am trying to figure out how to solve:
$\min_U r_{p}$
where $r_{p}=\alpha^\intercal\omega$ and $U$ is a sphere centered at $\alpha$ with radius equal to $\chi|\alpha|$ . ( $\omega$ is a vector or ...
0
votes
1
answer
153
views
Difference of two optimization problem's optimal value
Let we have two following optimization problems:
\begin{align}
\text{(P1)}\quad \alpha_1 = \max_{x_1,\ldots,x_M} &\quad \sum_{m=1}^{M}\log(1+f_m(x_1,\ldots,x_M))\\
\textrm{s.t.} &\quad \...
- 287
-1
votes
2
answers
487
views
Any efficient software/package/toolbox for nonlinear programming for MATLAB
I want to solve a nonlinear programming problem with the objective function being coded as a recursive function in Matlab. I have tried “fmincon”, but it could not get the solution due to large number ...
- 1
-1
votes
1
answer
328
views
About the critical points of quasi-convex functions
What do we know about the structure of critical points of quasi-convex functions?
I am looking for statements like "the critical points of a quasi-convex function are always either a global minima ...
- 2,246
-1
votes
2
answers
775
views
Is finding a local minimizer of a NP-hard optimization problem is still NP-hard [closed]
I was wondering if for a NP-hard optimization problem, I only want to find its local minimizer, is it still NP-hard or NP-hard is only true when trying to find a global minimizer?
- 11
-1
votes
1
answer
99
views
Existence of continuous selection for metric projection
Let $(X,d)$ be a separable complete geodesic metric space and let $K$ be a compact (non-empty) subset of $X$. Without assuming things like linearity, the convexity of $K$, and locally convexity, ...
-1
votes
1
answer
768
views
How can I deploy a CG-Steihaug algorithm for trust region sub-problem solving?
I'm studying various optimization methods and on this occasion, I'm trying to tackle the Trust Region Problem by solving the sub-region problem with the Steihaug-CG algorithm in Python. I'm using the ...
-1
votes
1
answer
71
views
Compute the proximal of a mapping [closed]
Let $ f: \mathbb{R}\longrightarrow \mathbb{R}$: compute
proximal of following mapping
$$ f(x)= \sqrt {1-x^2} $$
for $ x \geq 0 $
I know that the proximal is given by
$$ \operatorname{prox}_{\!...
- 1
-2
votes
1
answer
141
views
Prove the function $g(x,y,t)\ge1$
I have the function
$$
g(x,y,t)=\frac{(8x^2y^2+f_+(x,y,t)-\cos(2t))(8x^2y^2(1+(x+y)^2)+(x+y)^2(f_-(x,y,t)-\cos(t))+4xy(x+y)\sin(2t))}{64x^4y^4(1+(x+y)^2)}
$$
with
$$
f_{\pm}(x,y,t) = 1+2x^2+2y^2\pm4xy\...
- 375
-2
votes
1
answer
183
views
Property of positive semi-definite
Let $A$ is a positive semi-definite matrix like this:
$$ A = \begin{bmatrix}
1 & \alpha_{1,2} & \alpha_{1,3} & \alpha_{1,4}\\
\alpha_{1,2} & 1 & \alpha_{2,3} & \alpha_{2,4}\\
\...
- 25