A non-linear optimization problem includes an objective function (to be minimized or maximized) and some number of equality and/or inequality constraints where the objective or some of the constraints are non-linear. Use this tag for questions related to the theory of solving such problems or for ...

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Nonlinear least squares problems with binary variables

I want to solve the heat equation $T_t(x,t) = - L_x . T(x,t) + F(x,t)$ in an edge-weighted graph where $L_x = \sum_i x_i e_{ij}$ is weighted Laplacian matrix of the graph. Then I conclude to the ...
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27 views

Convexity Proof with constraints on the gradient

Consider a minimization problem $(P)$ : minimize $f(x)$ subject to $\delta_C(x) \leq 0$ Now assume that $\emptyset \neq C \subset \mathbb{R}^n$ is convex and let $f: \mathbb{R}^n \to \mathbb{R}$ be ...
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38 views

About the alternating optimization

The problem is defined as follows: $$ min_{A,B,C} f(A,B,C) $$ and the problem couldn't solve by gradient descent or close-form solution. Thus, the usual way is to use the alternating optimization: ...
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1answer
37 views

Convexity proof - can I get some pointers?

Prove that $C \subset \mathbb{R}^n$ is convex iff $\forall m \in \mathbb{N}$ and every set of $m$ points $\{x_1,...,x_m\} \subset C$ we have that $\sum_{i=1}^m \lambda_i x_i \in C$ Where ...
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17 views

How to solve a multidimensional stochastic optimization program with minimizing operators?

How to solve the following optimization program? $$ \max_{u_i^h,u_i^l,i=1,2}\Pi =E\big[a\min\{u_1^h,u_2^h, K_1^h,K_2^h\}+b\min\{u_1^h,u_2^l,K_1^h,K_2^l\}\\ ...
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1answer
113 views

Newton optimization algorithm with non-positive definite Hessian

In the newton optimization algorithm to find the local minimum $x^*$ of a non-linear function $f(x)$ with iteration sequence of $x_0 \rightarrow x_1 \rightarrow x_2 ... \rightarrow x^*$ all $\nabla ^2 ...
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1answer
121 views

Optimization of Frobenius Norm and Nuclear Norm

How to solve the following optimization problem, \begin{equation} \boldsymbol{\hat{x}} = argmin_{\boldsymbol{x}} \frac{1}{2} \| \boldsymbol{x - y} \|_F^2 + \lambda \| \boldsymbol{x} \|_{*} ...
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Difficulty in understanding a solution: Constraint minimization of sum of Non-symmetric matrices

I am trying to understand why there is significance difference in the performance of two proposed solutions. Original question (Constraint minimization of sum of Non-symmetric matrices) ...
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13 views

How to minimize $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $

I'm new in optimizations and i am trying to understand how to obtain $ v $ that minimizes $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $ where $\rho(x)$ - continuous ...
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1answer
46 views

Find $\min x^TAy+b^Tx+c^Ty$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$

The problem seems to be easy but I can't find a solution :( Problem: Given $A\in\mathbb{R}^{m\times n}, A\ge 0, b\in\mathbb{R}^{m}, c\in\mathbb{R}^{n}$. Minimize $f(x,y) = x^TAy+b^Tx+c^Ty$ subject to ...
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37 views

how to find the solution of this cost function?

I have the following cost function. $J = \sum_{i=1}^N a\, Trace(W^TX_iW) - b\, Trace(W^TY_iW)$ Where $X_i$ and $Y_i$ are symmetric matrices, $a$ and $b$ are scalars. How can I find W?
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2answers
107 views

Constraint minimization of sum of Non-symmetric matrices

I am trying to find closed form solution to following problem \begin{equation} \begin{array}{c} \text{min} \hspace{4mm} \big(\lambda_1\left( \mathbf{y}^T V^{(1)}\mathbf{x} \right)^2 + ...
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17 views

Local optimization with multiple starting values \approx global optimization?

I need to find the minimum/maximum of a nonlinear function but the constraints in the optimization problem make it tougher to solve (not a convex problem). I don't have a good global optimization ...
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17 views

Finite difference method hessian matrix

In the finite difference method, after every iteration, do you have to update the hessian matrix and the grad function?
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57 views

Determining active constraints in KKT

Suppose there is a constrained optimization problem having inequality constraints. We can solve it using Karush-Kuhn-Tucker conditions. My question is how do we determine which constraints are active ...
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19 views

maintaining monotonicity in an optimization problem

I have written a program for optimizing a set of generators. I have hourly price and cost data and need to figure out when a generator should run or just stay off. I describe the problem in more ...
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1answer
25 views

Problems with vector vector derivative in optimization

I have a loss function of the followoing form: $L(\mathbf{a}) = \|\mathbf{b} - \mathbf{a}\|_2^2$ Where, $\mathbf{a}$ and $\mathbf{b}$ are vectors of dimension $d\times 1$. I need to calculate ...
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25 views

Convexity over a line given a convex interval [duplicate]

Let $f : \mathbb{R}^n \to \mathbb{R}_∞$ be a function. I want to prove that $f$ is convex over the line $L_{v,x_0}$ iff $\psi : \mathbb{R} \to \mathbb{R}_∞$ $\psi(t) := f (x_0 + tv)$, is convex ...
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41 views

Name of method which includes Taylor linearization inside fixed point iteration

I read paper about Horn-Schunck multiscale method for computing optical flow Core part of this algorithm is minimizing some functional. One part of functional contains nonlinear term inside L2 norm. ...
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1answer
43 views

understanding a statement in Gill, Murray and Wright “Practical Optimization”

Hi: I'm reading the book "Practical Optimization" and there's a part in Chapter 3 that I can't prove to myself but I'm sure it's true. On page 64, they define the Taylor expansion of $F$ about ...
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23 views

Analytic minimization of linear algebraic expression with nonlinear constraint

I'm trying to solve the problem of minimizing $a x_1 + (1-a) x_2$ with constraint $b - b^2 \sqrt{(1-x_1)(1-x_2)} - \sqrt{x_1x_2}=0$ and where all variables lie in $(0,1)$. It comes out of a physics ...
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33 views

Efficient method for refining parameters in nonlinear curve fitting

I have time-series electrical current data $i(t)$ with transient steps in it which are convoluted with the hardware filter used in data acquisition. As a result, the real steps in current, which would ...
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1answer
51 views

fastest path between two points with current

This question is an extension of the following trivial problem: on a stationary body of water, a motorized device is 100m S and 10m E of a buoy. Given the device can move 5m/s in stationary water, ...
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17 views

Optimal design for constrained Bayesian slope intercept model

Here is a problem I've been stuck on for quite a while. Consider the model \begin{equation} \mathbf{y}=\mathbf{H}\pmb{ \theta }+\pmb{\epsilon }. \end{equation} The design matrix is given by: ...
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1answer
59 views

Is there any algorithm for finding the minimum distance to the complement of a convex set?

There have been some algorithms for finding the projection from a given point onto a convex set. This problem seems to be quit easy because of the convexity of the set. However, in the case of finding ...
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34 views

solver for non-convex matrix optimization with convex constraints

So here is the problem: $\max_{D} ~~ \|A+BD\|$ subject to $\|D\|<1$ (any norm you like) where matrices A and B are given. The cost function is evidently convex as well as the constraint, but ...
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51 views

Solution of a linearly constrained quadratic programming problem

What is the solution of the following optimization problem: \begin{align} &\min{\mathbf{p}^\mathrm{T} \mathbf{B} \mathbf{p}}\\ &\text{subject to}: \mathbf{0}\leq{\mathbf{p}}\leq \mathbf{1}. ...
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49 views

Maximizing a convex function outside a convex set?

I want to prove the following equality: \begin{equation*} \min_{x: x^2 \ge t - x} x^2 = \max_{0 \le \mu \le 1} \left( \mu t + \frac{\mu^2}{4 (\mu -1)} \right). \end{equation*} The objective function ...
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Nonlinear optimization with range constraint

I am trying to solve the following problem, but so far each time trying to program either one or more of the constraints are not satisfied (using Matlab). Phrasing the problem in two different ways, ...
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59 views

Is there any way to justify this approximation of the solution of a rather simple equation?

I am considering the problem of finding the root of equation $$f(x)=-x+\sum_{i=m}^{i=n} \sqrt{x+i}=0$$ where $m,n$ can both be from very small to very large integers. Since, in a single calculation, ...
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139 views

An Interesting Resource Allocation Problem

Here is the problem: \begin{array}{ll} \text{minimize} & \sum_{i=1}^N \frac{1}{1 + \textrm{exp}(C_i + x_i)}\\ \text{subject to} & \sum_{i=1}^N x_i \le R \\ & x_i \ge 0, ~ i = 1,2,...,N ...
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316 views

Bidding Tic Tac Toe

In regular tic tac toe, both the players get alternate chances. This is a variant of that. Player $A$ has $\$x$ amount and player $B$ has $\$y$ amount as initial balance. Assume that $y>x$. Both ...
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62 views

Nonlinear non-convex semi-infinite programming with norm equality constraint

In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a ...
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1answer
54 views

Optimization problem with an added quadratic inequality constraint

Consider the following (non-convex) optimization problem on the real variables $\lambda_\ell^\pm$ with $\ell=1,\ldots,n$ \begin{align} \mbox{maximize}&\quad ...
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1answer
87 views

What is the motivation behind the, convex and concave closures of submodular functions?

What is the motivation behind the , convex and concave closures of submodular functions? Also, my understanding is that the submodularity condition is somewhat like concavity which makes it counter ...
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1answer
155 views

Scale ellipsoid maximally within polyhedron

Given an ellipsoid around the origin with scaling parameter $e$ in the form $x^T E x \leq e$ and a polyhedron $P$ given by $A x \leq b$, how can we define an optimization problem that maximizes e such ...
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26 views

How to interpolate a function with a reproducing kernel

I am trying to interpolate a function that is noisy, but I know with a high amount of certainty about a third of the points in the series. I am trying to estimate the smooth mean of the signal via a ...
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33 views

Solving an inverse squared sum

How would I go about solving this sum for $x$? $$\sum_i\frac{a_i}{(x+b_i)^2}=C$$ Where $\mathbf{a}$ and $\mathbf{b}$ are vectors and $C$ is a constant, and $x$ is a single number. It's for an ...
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1answer
260 views

Gauss-Newton vs Gradient descent

I would like to ask first if the second order gradient descent method is the same as the Gauss-Newton method. There is something I didn't understand. I read that with the Newton's method the step we ...
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1answer
34 views

Finding sparsest solution of a linear system

I want to find the solution $x$ with most zeros in its components, to: $Ax=b$ for $A\in \mathbb{R}^{k \times n}, b \in \mathbb{R}^k$ ($k < n$), where $x \in \mathbb{R}^n$ has no additional ...
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119 views

Strong duality for nonconvex quadratic program (with multiple constraints)

Consider the following optimization \begin{eqnarray} P_1: \quad &\underset{x\in\mathbb{C}^N}{\mathrm{minimize}}&\; f_0(x) \\ &\mathrm{subject\;to}&\; f_i(x) \leq 0, i=1,\ldots,m \\ ...
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Structural / design / meta optimization - is there mathematical theory. Optimization over categories?

There is huge branch of mathematical optimization theory, but it mostly considers the finding optimal parameter values for the predefined structures. There are variational calculus and optimal control ...
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102 views

From constrained to unconstrained maximization problem

I have the following constrained maximization problem $$ \max_{X_1,X_2,...,X_i,...,X_N} \sum_{i=1}^{N}X_i f_i(X_1,...,X_N) \hspace{0.2 cm} \text{subject to} \sum_{i=1}^{N}X_i-B\leq 0 \text{ and } ...
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61 views

short-sale constraint with nonpositive-definite matrix in portfolio optimization

This question is about portfolio optimization in R. I have a nonpositive-definite matrix. I have handled with the singularity. Unfortunately, quadprog etc. optimization packages fail to solve the ...
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1answer
49 views

Minimizing a function in Mathematica

Edit: I simplified the function using $\textbf{Simplify[...]}$ How can I minimize this function of $x$, where $l$ is a positive constant? $$\frac{1}{2} \sqrt{\frac{x}{l}+\frac{l}{x}+4 x^2-2}$$ ...
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27 views

Solving nonlinear system of ODEs

I have the following system of differential equations: $$ \begin{cases} \frac{dx}{dt} = (1 - y) x - 0.4 xu \\ \frac{dy}{dt} = (x - 1)y - 0.2yu \\ \psi_1' = - \frac{dH}{dx} = (-1 + 0.4u)\psi_1 + y ...
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1answer
31 views

Constrained nonlinear optimizacion

I have encountered this optimization problem while trying to implement the method proposed in this very interesting paper: http://www.mae.cuhk.edu.hk/~cwang/pubs/JCISERealTimeSkeleton.pdf the ...
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38 views

What does coordinate descent actually do?

We've done a bunch of theoretical stuff in my optimization class, but basically no time for the actual implementation details. I'm trying to get an understanding of coordinate descent, which if I'm ...
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53 views

Linear optimization w/ linear and non-linear inequality constraints

Given dependent variables $Q_i$ and independent variables $x_i$, $y_i$, $z_i$ where $i=1,⋯, N $ which are related via the following system of N linear equations with parameters $P_1$, $P_2$ and ...
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97 views

If $2000 m^{2}$ of material is used to to construct a box…,then what is the largest possible volume of the box?

If $2000 m^{2}$ of material is used to to construct a rectangular box with a square base and an open top,then what is the largest possible volume of the box? I put an equation for the volume : $V = ...