0
votes
0answers
21 views

minimizing sum of squares

Say i have the following optimization problem. min $\sum\limits_{i=1}^m \parallel r - (y_i - Rx_i) \parallel_2^2$. where we are optimizing over $r \in R^n$ and also $R \in R^{n, n}$ is given. Also, ...
1
vote
0answers
19 views

Inequality optimization, KKT condition.

So we have the problem: maximize $x^2+y^2$ subject to $x^2-y \leq3$ and $y\leq 1$. And I sorted out the KKT conditions for the problem (is here where the problem is?): $2x=\lambda _12x$, ...
5
votes
3answers
95 views

Find min of $\frac{\left( \sum kx_k \right) \left( \sum x^2_k \right)}{\left( \sum x_k \right)^3}$

With $n \ge 2$ and $x_1,\ x_2,\ \dots,\ x_n > 0$. Find the minimum of: $$ M = \frac{(x_1 + 2 x_2 + ...+ nx_n)( x^2_1 + x^2_2 +...+x^2_n)} {\left( x_1 + x_2 +...+ x_n \right)^3}$$ For specific ...
1
vote
3answers
42 views

Minimizing sum of squared distances from point to spheres

Given some spheres with known radius and known origin in three dimensional space, I want to find the point P that lies "closest" to all these spheres. The meassure of closeness, I guess, will be the ...
0
votes
1answer
15 views

Finding the maximum/minimum of a homogeneous function on $R^n$

Suppose that $f:R^n\to R$ is homogeneous. Also, suppose that the $argmin_xf(x)$ is non-empty. Is it true that if there exist $x^*\in R^n$ such that $f(x^*)=0$, then $x^*=argmin_xf(x)$?
1
vote
1answer
37 views

KKT point of a constrained optimization problem

Min$_{x}~x$ Subject to $x \geq 0$ For this problem, is $(x^{*}, \lambda^{*})=$$(0,0)$ a KKT point ? My try : I formulated corresponding Lagrangian and tried to find out the KKT point(s). ...
0
votes
1answer
32 views

Convex functions -> quasi-convex functions -> … can we weaken the assumptions?

First of all let me say that I'm new to optimization. I realized that quasi-convex functions share with convex functions some nice properties, so I wonder if we can push the weakening a little ...
0
votes
0answers
31 views

least-square optimization with linearly depend solution $x$

What is the exact solution $x_{n \times 1}$ of the following constrained optimization problem \begin{align*} &\min \|AX-b\|^2 \\ s.t.&Cx=0 \end{align*} where $A$ is full column rank $m \times ...
1
vote
0answers
33 views

An indirect optimizing problem

we have $z_1=x_1y_1$ , $z_2=x_2y_2$ and $z=xy$. we know $x<x_1+x_2$ and $y<y_1+y_2$. Can we conclude minimizing $z_1$ and $z_2$ will lead us to minimum (or lesser values) of $z$? ...
1
vote
2answers
30 views

Minimizing a convex cost function

I'm reviewing basic techniques in optimization and I'm stuck on the following. We aim to minimize the cost function $$f(x_1,x_2) = \frac{1}{2n} \sum_{k=1}^n \left(\cos\left(\frac{\pi k}{n}\right) x_1 ...
0
votes
1answer
32 views

Can SVD help to solve (inequality) constrained least squares problem?

Consider the following minimization problem: $$ ||Q u - h^{o} ||^{2} \to min \;\;\; s.t. \; u \geq 0 $$ where $Q$ is $m \times n$ matrix and $u$ is $n$-dimensional vector and $h^{0}$ is ...
0
votes
1answer
19 views

Non-linear estimate parameter

I have one non-linear function that define $$E_x(a,b)=\int K_\sigma(y-x) \cdot(b-b. e^{-a\cdot f(y)} \,) dy$$ where $y$ is neighboor points of $x$; $f(y)$ is a function of $y$; and $a$ is constant. ...
0
votes
0answers
5 views

Linear Quadratic Bilevel Programming Problem

How to solve this type of linear-quadratic bilevel programming problem ? Please help.
0
votes
0answers
42 views

Expressing rank condition of a matrix in terms of its elements

Let $x \in \mathbb{R}^{n}$, define $X = xx^{T}$. I have an optimization problem with some linear constraints and few quadratic constraints, and I have to solve for $x$. Using $X$ as the unknown ...
1
vote
0answers
32 views

using lsqcurvefit to fit piece-wise linear

I would like to use this function to fit piece-wise linearly to a set of data. Namely, I want to fit them with several linear segments. Including other requirements, I would not want the segments ...
2
votes
0answers
54 views

Maximize a function subject to the constraint $x^2+y^2=R^2$

Please help me how to deal with maximization of function $$f(x,y)=1-e^{-\pi x}+e^{\pi x}\left[1-\cos(\pi y)+\sin(\pi y)\right]$$ subject to the constraint $x^2+y^2=R^2$. Using Lagrange ...
1
vote
1answer
53 views

Why is this a quadratic programming problem?

I am sorry if this is a stupid question, I'm very new. How would I minimize the following objective? $\sum_{k=1}^p\| I_{k} - M_{k}A \|^2$ Each I and M are known. I am told I can use a quadratic ...
3
votes
1answer
41 views

How to find the minimal value of this function under such constraint

$f(x,y)=x-\sqrt{y-x^2}$ with $a<x<b$ and $x^2<y<c*x-d$. What I did is, first take partial derivative at $x$ and $y$ respectively, however, there is no critical point because fy is always ...
2
votes
1answer
74 views

Extremum of functional of a complex function

consider functional $E$ defined by $$E[z]=\int F(x,z(x))dx$$ where $F$ is a complex-valued nonlinear function. How can we find the function $z(x)$ so that $$G=|E|^2=EE^*=\iint ...
1
vote
1answer
31 views

The most optimal way to solve this set of non-linear equations in high dimensions

So I have a series of non-linear equations which I wish to solve as fast as possible, to illustrate for the case of $n = 4$, I have the following equations: \begin{gather*} ...
0
votes
0answers
30 views

Combinations of Convex Functions

I'm looking at the following non-linear optimization theory problem: Let $\gamma$ be a monotone nondecreasing function of a single variable (that is, $\gamma(r) \le \gamma(r')$ for $r' > r$), ...
0
votes
0answers
36 views

Proofs involving closedness of compositions of mappings.

I've feel like I've gotten myself in over my head in a non-linear optimization course I seem to lack the mathematical maturity for(I'm an undergrad, I've taken the calc series, Intro to differential ...
0
votes
0answers
11 views

Implementing a projection with KL-divergence

I want to implement the following and I am looking for an easy/fast way to implement it(the programming language does not matter). Assume that $p(\mathbf{x})$ is a proper probability distribution and ...
2
votes
1answer
31 views

Conceptual Understanding of Non-Linear Optimization Problem

I'm in non-linear optimization, and I'm having trouble wrapping my head around what this problem is asking me for. If anyone could help with a conceptual explanation (not an answer!), it'd be greatly ...
0
votes
0answers
20 views

complexity of an optimization problem

Consider $n$ variables $x_1, \cdots, x_n$ with the constraint $\sum_{i=1}^n x_i=1$ and $x_i\geq 0$. I want to minimize $\vec{a}^T (I-\alpha A(\vec{x}))^{-1} \vec{b}$, where $\vec{a}$ and ...
1
vote
0answers
30 views

How to minimise an objective function which is not a direct function of the decision variable?

I have a problem with partitioning a water network by closing some pipes. I use some graph theory techniques to find some candidate pipes to close; but to select which pipes among them to close (my ...
1
vote
1answer
31 views

Prove that a multivariable function doesn't have global extremes

So my question is actually this. Say I have a function $F:\mathbb R^2\to\mathbb R$. If I find all the potential local extremes by finding the roots of the partial derivatives and I find that only one ...
-1
votes
2answers
36 views

Constraints in optimization; redundant hardness?

This is not an accurate mathematical problem, and rather a philosophical and ambitious question. As far as I know, unconstrained problems are easier than constrained problems; right? This is mostly ...
3
votes
1answer
50 views

Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
1
vote
0answers
24 views

Effect of approximating a non-differentiable function on optimisation of minimisation

I am looking at a problem of constrained minimization, where the function to be minimized contains the Heaviside function, and as such is not twice continuously differentiable. My question is what ...
1
vote
0answers
27 views

Representing a 2D function as a sum of rectangles of arbitrary shape and orientation

Suppose I am given a non-negative function $f(x,y)$ defined for $x \in [0,1]$ and $y \in [0,1]$. I'd like to represent this function as a weighted sum $w_i$ of a small number of rectangular apertures. ...
2
votes
1answer
41 views

Initialization of Limited-memory BFGS (using libLBFGS)

I am using the package libLBFGS in order to minimize an objective function, for which the first derivative (with respect to the optimization variable) is known and computable. I use the default ...
2
votes
2answers
107 views

Are solutions to optimization problems with smooth, continuous, and strictly concave objective functions and linear constraints always unique?

As an example, if I have a minimzation problem where my objective function is represented by a sphere in n dimensions (one dimension per decision variable), and all my constraints are linear, then ...
1
vote
1answer
63 views

First-order necessary condition for relative minimum point

I'm studying linear and nonlinear programming and I came across with the following proposition : given $\rm x\in\Omega$ we are motivated to say that a vector $\mathbf d$ is a feasible direction at ...
1
vote
1answer
27 views

A non-linear optimization problem

I have the following optimization problem on the variables $a_1, ..., a_n$: $$ minimize \frac{\sum_{k=1}^{n}\max(k\cdot a_{k},1)}{\sum_{k=1}^{n}a_{k}} $$ $$ such\ that\ \ 0\leq a_k\leq 1\ \ \ (k=1, ...
1
vote
0answers
176 views

How to solve nonlinear constrained optimization in Matlab?

I have to solve a nonlinear constrained function in matlab, and I am not familiar with it's commands. the problem is: minimize $E(b,c)$ constraints: $k1< c\sqrt{b}< k2 ; c/6>k3$ Note: E(b,c) ...
2
votes
1answer
144 views

Prove or disprove the conjecture about the function below.

After thousands of numerical tests we stated the conjecture that their is exactly one local extremum of the function below. $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + ...
0
votes
5answers
123 views

How to find the minimum of the function?

How to find the minimum of the following function $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + {\rm e}^{-x_{i}\,w}} -y_{i}\right)^{2} $$ where $x_{i}, y_{i} \in \left(0, ...
1
vote
1answer
104 views

Definition of tangent cone in continuous optimization .

Looking at the definition of tangent cone in continuous optimization : If $M$ is a open subset of $\mathbb R^n$ $x \in M$, The tangent cone of $M$ at $x$ is defined by $$\mathbb T (M, x) = \big\{d ...
1
vote
1answer
58 views

Minimizing a linear function on a strictly convex set.

All the theorems that I know considering the uniqueness of a solution to a minimization/maximization problem requires the strict convexity/strict concavity of the objective function. But consider the ...
1
vote
1answer
58 views

Third and higher order optimality conditions?

In the derivation of first and second order optimality criteria for a vector $X^*$ to be a local optimum to an unconstrained problem, we ignore the higher order terms of Taylor's expansion as we ...
0
votes
0answers
40 views

How do I find out if a critical point of a function is a maximum or a minimum?

If I've found the critical point of a function defined in some constraint (perhaps using Lagrange multipliers and the like); how do I find out if it's a relative/global maximum/minimum of a function ...
0
votes
0answers
27 views

Defining a metric for non-convexity of functions

I have a non-convex function of two variables, $f(x,y)$. I drew it and realized that the function has a convex shape except in very few points where the non-convexity is very minor. I am looking for ...
0
votes
0answers
35 views

Using Newton's method to find an optimized matrix

I'm trying to apply Newton's method to find a local optimum of the matrix $\Sigma$ to minimize the objective function: $f(\Sigma) = -\sum_{n=1}^{N}\left(-\ln{2\pi} - ...
1
vote
0answers
28 views

Decomposition of a symmetric semi-definite matrix into sums of sparse symmetric semi-definite matrix

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 ...
4
votes
2answers
79 views

Maximize $\prod\limits_{i=1}^n m_i$

Someone visits a market where one hundred different types of fruit are sold. All types cost $1$ euro per pound. The utility the buyer receives from buying $m_1$ pounds of the first type of fruit he ...
0
votes
0answers
46 views

KKT minimization problem

Solve $x^2 - 2y \rightarrow \min$ subject to $\max\{3x^2, e^y + 2\} + \sqrt{x^2 + y^2 - 2x + 1} \leq 6x + \sqrt{5}$ and $ \sqrt{x^2 + y^2 - 4x - 4y +8} -2x+2y \leq 0$ I tried computing the ...
0
votes
1answer
95 views

KKT maximization problem

$x^2y \rightarrow$ max, such that $x^2 + 4xy \leq 1, x \geq 0$ and $y \geq 0$. I think I need to use the KKT conditions here. I did however not yet succeed in solving it, so could someone ...
0
votes
0answers
37 views

Simulated annealing with constraints

Say we have an objective function with many local optima, and we wish to find the global optimum. Further, assume the method of choice is simulated annealing. Now, suppose there is a constraint on the ...
0
votes
0answers
26 views

How fast Interior Point method can be when solving Quadratic Programming problems?

Given the following Quadratic Programming problem: $\;\;\;\;\; \min x^TQx+c^Tx $ s.t. $Ax=b$ $\;\;\;\;\; x\ge0$ where $x\in \mathbb{R}^n$, $Q \in \mathbb{R}^{n \times n}$ is a positive ...