Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.

**0**

votes

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19 views

### Bounding function of norms in constrained vector space

$v$ is a vector of length $n$, where $v_1 = 1$ and every element $v_i \in [0,1]$
$w = \| v \|_1^1$
$x = \| v \|_2^2$
$y = \| v \|_3^3$
$z = \| v \|_4^4$
Can you recommend a strategy for achieving a ...

**2**

votes

**0**answers

30 views

### Maximizing a convex bounded function of a PSD matrix

Let $f(X)$ be convex and continuous function , with $X$ a PSD matrix.
Assume that under the affine set of constraints $\mathcal{A}(X)=b$ and the convex constraint $f(X)\le1$ there is an optimal, ...

**4**

votes

**0**answers

87 views

### maximize non-convex composite function

I want to maximize a composite function over a convex set
\begin{equation}
\begin{aligned}
& \underset{\mathbf{p}}{\text{maximize}}
& & f(\mathbf{p})-g(\mathbf{p})\\
& \text{subject ...

**3**

votes

**1**answer

68 views

### Constrained optimization (QCLP) over $x$ with the constraint $x = Az$

I have a problem that looks very much like a (norm-constrained) linear program, but with an extra constraint that is unusual for me. The problem is, given a matrix $A$ and a vector $w$,
$$ \min_{x ...

**0**

votes

**0**answers

124 views

### Asymptotics to Taylor expansions?

I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments.
Maybe you guys can help.
...

**2**

votes

**1**answer

129 views

### Can the nonlinear matrix equation $Bx=p+sgn(x)$ have an explicit solution?

Given $n > 0$, $0 \leq i \leq n$ is an integer, $D = diag(d_1, \dots, d_n)$ is positive definite, $e_i$ is the $i$th column of a $n \times n$ identity matrix, $u \in R^n$ such that $B = D + u * ...

**0**

votes

**1**answer

47 views

### How to compute the direction of slowest ascent from the minimum of a strongly convex function?

Consider a twice differentiable strongly convex function $f:\mathbb{R}^n \rightarrow \mathbb{R^+}$ that attains its minimum value at the point $x^*$. I am wondering if one can compute a direction of ...

**3**

votes

**1**answer

190 views

### Valid KKT Constraint Qualification? Linear constraints not full rank, Jacobian of nonlinear constraints full rank and independ. of linear constraints

For a nonlinear optimization problem having only linear constraints, by the Linearity Constraint Qualification, no further constraint qualification is required for the Karush-Kuhn-Tucker (KKT) ...

**1**

vote

**1**answer

89 views

### Nonlinear least square with quadratic equality constraint

I am looking for an appropriate method or hint to solve the following constrained nonlinear least square problem:
$\operatorname{argmin}_X \sum_{i\in I} \|\mathbf{X}_i - \mathbf{X}_{i+1}\|_2^2 + ...

**2**

votes

**1**answer

43 views

### Calculating the Upper Bound on the Sphere Radius of Knotted Channel Surfaces

This question is motivated by trying to determine the upper bound on the thickness of a rope of fixed length (w.l.o.g. $2\pi$), with which a knot of given topology can be realized under the further ...

**4**

votes

**0**answers

34 views

### How does one go from convexity to submodularity?

If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular?
It would be helpful is someone can share some specific ...

**5**

votes

**2**answers

71 views

### convert a special case of nonlinear fractional programming into a convex problem

Is it possible to convert a fractional problem (maximization) with objective function equal to the ratio of a concave function and convex function ? This question sound impossible but I have read this ...

**0**

votes

**2**answers

113 views

### Estimating the shift in the $\lambda_{max}$ of a matrix under a diagonal perturbation

Given a matrix $A$ and a diagonal matrix $D$, what ways do we have to estimate, $\lambda_{max}(A+D) - \lambda_{max}(A)$? (Feel free to make other assumptions about the matrices that they are all ...

**0**

votes

**0**answers

57 views

### Is there a space in which the $\vec a$ in $\sin(a_1\cdot x)+\sin(a_2\cdot x)$ is linear?

Suppose one has equations of the form $\sin(a_1\cdot x_i)+\sin(a_2\cdot x_i)=y_i$ for $i=1...n$ (there are also amplitudes and phase shifts, but let's ignore those for now). I want to solve for $\vec ...

**1**

vote

**0**answers

44 views

### Optimization question: maximize quadratic objective with semidefinite constraints

I recently encountered the following optimization problem:
$\max \|AX\|_F^2$
subject to: $X\succeq0$ and $Xb_i\leq c_i$ for a collection of $T$ conditions: $i=1,\ldots,T$.
Matrices $A$ and $X$ are ...

**0**

votes

**0**answers

28 views

### Relaxation of non-convex QCQP with one quadratic and one linear constraint

According to Boyd we know that a non-convex QCQP problem with one quadratic constraint has strong duality with the relaxed SDP or Lagrange counterpart. (check "Convex Optimization" by Boyd, Appendix ...

**5**

votes

**1**answer

93 views

### Resource Constrained Routing with Refueling

What are good algorithms (resp. models) for calculating optimal or near optimal routes while taking into account fuel consumption, options for refueling and, limited tank capacity?
Especially modeling ...

**2**

votes

**1**answer

66 views

### Fixed point iteration on symmetric biconvex function

Suppose $X\subseteq\mathbb{R}^n$ is a convex set and that a function $g(x,y):X\times X\rightarrow\mathbb{R}_+$ is smooth, "strictly biconvex" (strictly convex in $x$ and $y$ independently but not ...

**0**

votes

**0**answers

34 views

### An obstacle problem

Let $f:[0,T]\to \mathbb{R}$ be an increasing function with $f(0)=0$. We want to maximize $f(T)$ with the following constraints:
$|f^\prime(t)|\le M,\quad \forall t\in[0,T]$
$f(t)\le g(t),\quad ...

**2**

votes

**1**answer

163 views

### How can I find the maximum value of this function?

For given values of $A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^m$, how can I find the value of:
$$
\max_{x \in [0,1]^n} \|Ax+b \|_1
$$
Or is this problem NP-hard?

**3**

votes

**0**answers

78 views

### Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same

I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, ...

**5**

votes

**2**answers

75 views

### Discrete optimization problem

Suppose you had $N$ many fixed points $X_1, X_2, ..., X_N$ in some Euclidean space $R^d$ and from these coordinates you had to choose $n$ many of them ($n \leq N$ also being fixed) to form a subset ...

**1**

vote

**0**answers

117 views

### Why hexagons? The maximal minimum of a sum of cosines on the plane with frequencies on the unit circle

We wish to maximize the minimum of a weighted sum of cosines in the plane, when the frequency components are on the unit circle. Formally:
$$\max_{\{ a_i,\theta_i,\phi_i \}_{i=1}^{N} } \min_{(x,y) ...

**1**

vote

**1**answer

138 views

### Maximal minimum for a sum of two (or more) cosines

Please prove (or disprove, and give the correct answer):
$$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right]
$$
In other words, find $r \geq 1$, ...

**1**

vote

**0**answers

37 views

### Frobenius nearest non-negative Gram matrix of balanced row-sums

Let $W \in \mathbb{R}^{n \times n}$ be any non-negative real symmetric matrix. For $k \leq n$, let $\mathcal{F} := \{X \in \mathbb{R}^{n \times k} \ | \ X \geq 0, X \mathbf{1} = \alpha \mathbf{1}, ...

**3**

votes

**2**answers

176 views

### Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices

I'll first provide the background.
Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables
$x^{(1)},\ldots,x^{(n)}$.
We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...

**0**

votes

**0**answers

104 views

### Global and local maxima in a weighted sum of logarithms of linear functionals?

Constrained Optimization Problem
Is is possible to describe, and locate efficiently, the maxima of the function $f$, as described below in the parameters $\mathbf{x} = (x_1,...,x_m)$. The constraints ...

**0**

votes

**0**answers

70 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, ...

**2**

votes

**1**answer

93 views

### Rate of convergence for cyclic gradient descent

I'm trying to solve the optimization problem
$\min_x \frac{1}{n} \sum_{i=1}^n f_i(x)$
where $f_i$ are (strongly) convex, smooth, lower semi-continuous, etc.
However, I am not able to do conventional ...

**5**

votes

**2**answers

544 views

### Find minimum-area ellipse which encloses two ellipses

I need an efficient algorithm to find the ellipse with the smallest possible area which encloses two given ellipses. The given ellipses are constrained to have coincident centers at the origin but can ...

**2**

votes

**0**answers

212 views

### Find minimum-area ellipse enclosing a set of ellipses, all centered at the origin

Given a set of N > 2 (two-dimensional and coplanar) ellipses, all centered at the origin, how do I find the ellipse with the minimum area which encloses all of them?
Background:
Thanks to Will Jagy ...

**3**

votes

**1**answer

205 views

### Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...

**5**

votes

**0**answers

172 views

### Characterizing matrices with rank constraint

Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M]=\{Q\in\Bbb\{0,1\}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb\{0,1\}^{n\times ...

**2**

votes

**0**answers

266 views

### Programming workbooks in C++ and Research Math [closed]

I know the basics of C++ by taking a few courses and going through "C++ Primer" by Lippman. As a math graduate student, I would love to get my hands on some programming-math exercises geared towards ...

**2**

votes

**1**answer

53 views

### When is a convex program continuous in its constraint vectors?

Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$
I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and ...

**1**

vote

**1**answer

167 views

### No Strong Duality In Spite of Slater's Condition

I was reading some course notes here.
On Page 8, it says:
Note that strong duality holds here (Slater's condition), but the
optimal value of the last problem is not necessarily the optimal
...

**3**

votes

**0**answers

118 views

### existence of optimal control

I'm looking for an existential result in optimal control for the following class of problems:
Given $T > 0$, $\bar x, \hat x\in\mathbb R^d$, an instantaneous cost function $c:\mathbb R^d\times ...

**1**

vote

**0**answers

30 views

### Minimizing sum of functions, while keeping their values non-negative

Suppose we have data $\mathbf{x}_i$, $i\in \{1, \ldots, K\}$, and we're trying to find parameters $\hat{\mathbf{\theta}}$ such that
$$\hat{\mathbf{\theta}} = \underset{\mathbf{\theta}} ...

**3**

votes

**1**answer

129 views

### Constrained vs Unconstrained Optimization

I'm currently working on an optimization problem with a linear objective with linear and nonlinear constraints, i'm facing difficulties reaching a good solution, so i was advised to move the nonlinear ...

**0**

votes

**0**answers

32 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 ...

**6**

votes

**1**answer

475 views

### Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has
$$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)}
+\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi,
$$
where $f$ is the ...

**0**

votes

**0**answers

115 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 ...

**4**

votes

**2**answers

121 views

### Convexity of a (non-symmetric) function of matrices

Let $f : H_{n\times n} (\mathbb{C}) \rightarrow \mathbb{R}$ be the function on Hermitian, positive semidefinite matrices $f(A) = \frac{M_i (A)}{\det(A)}$ where $M_i(A)$ is the determinant of the the ...

**4**

votes

**1**answer

84 views

### Analysis of first-order methods for constrained convex optimization with approximate oracles

In many first-order optimization methods an oracle is needed whose action enforces the constraint/regularizations. For example, in projected gradient descent, conditional gradient method, and proximal ...

**2**

votes

**1**answer

116 views

### Optimization over symmetric polynomials

Consider the following constraint satisfaction problem: Let $\alpha_1 , \ldots, \alpha_k \in \mathbb{R}$ be given as well as an error parameter $\epsilon$. Find $p_1, \ldots, p_n$ such that
(i) $0 ...

**1**

vote

**0**answers

57 views

### modern exposition of exact ground state of classical XY model or Ising model

What is the state of art technique in solving exact ground state of Heisenberg model, meaning minimization of the H terms (hamiltonian) provided infinite spin space?
...

**1**

vote

**0**answers

55 views

### Likelihood convexification

I am doing constrained vector optimization using a non-convex non-linear likelihood function. My problem is of the following form:
$$\begin{align*}\hat Q &= \underset{\vec Q}{\arg\min} -\log ...

**2**

votes

**0**answers

80 views

### How to solve the following generalized quadratic programming problem [closed]

I want to solve a generalized form of a quadratic programming problem
$$\min_x \left(\sqrt{x^TPx}+\sqrt{x^TQx}\right)^2+c^Tx$$,
$$\textrm{ s.t. } Ax\le b.$$ Here, $P$ and $Q$ are both positive ...

**0**

votes

**0**answers

67 views

### Specific optimization problem solution procedures

Is there a standard procedure to solve following two optimization problems?
$$\mathsf{Problem\mbox{ }I}:\mbox{ }\min_{A\in\{0,1\}^{n\times n}:rk(A)=r}\mbox{ }\max_{R,S\in\Bbb R^{n\times ...

**4**

votes

**2**answers

239 views

### Convexity of a function of matrices

Let $A$ be an $n\times n$ positive-definite matrix. Let $0<\lambda _1 \leq \lambda_2 \leq \lambda _3 \ldots \leq \lambda _n$ be the eigenvalues of $A$. Let $n\geq k\geq 1$. Is the function $f(A) = ...