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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27 views

Optimization problem: solving one implies solving the reverse?

I am looking to solve an optimazation problem $Maximize_{x} [A(x)]$ s.t. $B(x)\geq B_0$, where $B_0$ is a constant. If I solve this problem (i.e, finding the optimal $x^*$ that optmize while ...
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106 views

Who knows Krotov's Method in Optimal Control Theory

I'm finishing my PhD thesis about applications of optimal control theory in the field of energy harvesting. In the course of my PhD I dealt with different ways to compute optimal controls, and I found ...
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27 views

How to Find the Maximum of a Function Represented by a Back-Propagation Neural Network?

First, I train a standard feed-forward neural network over a training set of data points. I get an approximate function, say $F(x)$, represented implicitly by that neural network. 1. How do I ...
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1answer
34 views

Placing Circles Onto Lines For Optimality

Suppose you have a yet to be determined number of vertical lines with length 50 on which you'd like to place as many circles as you can. Each circle is 10 units in diameter and its outside edge must ...
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21 views

constrained optimization including sum of two upper incomplete gamma function in both fitness function and constraint

i'm trying to solve this constrained optimization problem the constraint is $$\zeta=\frac {\Gamma \left(M,\frac {\lambda}{\left |\sum_{i=1}^Nw_i*h_{ei} \right|^2 + \sum_{i=1}^N\left |w_i \right|^2} ...
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49 views

Solving non-linear equations in a chosen subspace

I'm trying to find the root $\mathbf{f(x)=0}$ to the following sets of equations $$ f_1(x,y,z) = x^\prime - \frac{x}{\sqrt{x^2+y^2+z^2}} = 0 \\ f_2(x,y,z) = y^\prime - \frac{y}{\sqrt{x^2+y^2+z^2}} = ...
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11 views

Non linear Optimization for resource allocation

I want to maximize the sum rate of a wireless system while maintaining fair allocation by using fairness constraint. $R_k$ is the rate for each user. I have set up my objective function as : Maximize ...
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43 views

How are the tolerances evaluated in fmincon? specific/complete mathematical formulations needed.

I'm currently studying the stopping criteria about fmincon using different algorithms and I'm wondering how are the tolerances are actually evaluated and compared in the built-in function fmincon. ...
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48 views

May be a trivial question regarding constrained optimization

Optimization problem is to find $x$>0 which $min \ \ L=\frac{A\left ( B(\frac{C}{Cx-B}+\frac{1}{x})+2C\log(\frac{B}{x}-C) \right )}{B^3}$ $s.t \ \ x\leq K $ Rewriting the objective ...
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50 views

Optimize $\max _{x_1,x_2,…,x_N} N , \text{ s.t.} \sum_{i=1}^N f(x_i) \le a$

$Is there general theory for solving optimization problem of the following kind \begin{align} &\max _{x_1,x_2,...,x_N} N \\ \text{ s.t.}& \sum_{i=1}^N f(x_i) \le a\\ &\sum_{i=1}^N g(x_i) ...
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51 views

How to find values of non linear equations / system and solve for given values

I'm trying to find the value for the variable phase in a equation / system if amp=0.5 and freq=2.5 (note: i'm looking for several different phase values given amp and freq but this is a small ...
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1answer
166 views

Duality theory and nonlinear optimization

I have been studying nonlinear optimization recently and have come across some results that I need clarification for. I will do my best to explain them in detail below, providing citations where ...
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85 views

Solving linear objective functions with linear and non linear constraints

Is it possible to use Matlab commands intlinprog and fmincon to solve a linear programming problem with a linear objective ...
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13 views

Consider $x = A[v]_+ + B[-v]_+$, under what conditions on A & B is there a solution v for every x

Consider the vector equality $$x = A[v]_+ + B[-v]_+$$ where $[v]_+$ is the elementwise rectifier function, i.e. $$[x]_+ = \begin{cases} x &\mbox{if } x > 0, \\ 0 & \mbox{otherwise}. ...
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1answer
34 views

How to solve this matrix equation? Does the solution have a closed-form?

I am trying to find solution $X \in SO(3)$ for this matrix equation. $$PX - XQ + YXZ = K$$ where matrices $P, Q, Y, Z, K \in\mathbb{R}^{3\times3}$ are known. At the first glance, it does not seem to ...
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1answer
55 views

When exactly are quadratic objective functions polynomial time solvable

I'm considering quadratic programming problems of the form: $$ \max x^tQx+Bx$$ subject to the linear constraint $$ Ax \le b $$ I read that if is the case that $$ x^tQx + Bx \ge 0 \ \forall x$$ or ...
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42 views

Does a zero duality gap imply global optimality?

Let's say we are given a nonlinear optimization primal problem (P). Suppose that the dual problem (D) to the primal optimization problem (P) achieves a zero duality gap with a solution to the primal ...
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1answer
20 views

One definition of strong convexity (from textbook of Prof. Bertsekas in 2015)

In strong convexity, there are a few definitions, one of them is: $f$ is strongly convex over $\mathcal{C}$ with coefficient $\sigma$ if $\forall x,y \in \mathcal{C}$ and all $\alpha \in [0,1]$, we ...
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1answer
32 views

Is it possible to make a linear reformulation?

The question is what to do when we have a product of the three variables, quite different in their nature. One is binary, the second is real, and the third is from a discrete set of rational numbers. ...
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1answer
22 views

Question about least square allocation of an amount to different buckets.

Suppose we have to allocate $x$ amount to $k$ desired amounts. Is there algorithm to do this that minimizes the squared distance between the actual $k$ allocated values and the $k$ desired amounts? ...
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39 views

Optimization problem involving semidefinite matrix variable that is constrained to be a tensor product

I would like to solve the following optimization problem. With scalar $R$ and nine (mutually orthogonal) $9$-dimensional column vectors $\vec v_i$ all given ($\vec v_i\!'$ is the row vector Hermitian ...
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1answer
64 views

Find the minimum of a function for only positive values of the vector variable

Let variable vector $\vec{q}$ of size $m\times1$, and its diagonal counterpart $m\times m$ matrix $Q=diag(\vec{q})$, for some $m\in\mathbb{N}$. Define fixed parameter $n\times1$ vectors $\vec{p}, ...
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84 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 ...
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1answer
42 views

A question about a proof in nonlinear programming book

I have a question about the proof of Proposition 1.2.1 (Stationarity of limit points for gradient methods) in the nonlinear programming book (2nd edition) by Bertsekas. At the beginning of the proof ...
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40 views

Distance between a plane and set of points

Suppose $m$ data points belonging to a class represented by matrix $A$. Therefore, the size of matrix $A$ is $m\times n$. In addition, suppose $w\cdot x + b=0$ be equation of a plane in ...
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108 views

If this problem is not unbounded, what's wrong in this dual derivation?

In a paper with 100 citation, Robust Support Vector Machine Training via Convex outlier Ablation, a convex relaxation is used. In this paper, a form of robust svm proposed: \begin{align} \min_{0\leq ...
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1answer
37 views

Function Optimization

Assume any recursive function like: (just for example, my rekursive function is just too big to write) $x_{n+1}=\frac{(x_{n}-3)^{5}x_{n}^{2}}{a\sqrt{x_{n}}}$ (or any other non-linear function) Is ...
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21 views

finite differences nonlinear least squares

I am facing to following nonlinear least-squares problem: $$\min_{u,\gamma} \frac{1}{1000} \int_{ \gamma(x,y)^2} + \int_{ [u(x,y)−u (x,y)]^2} + \int_{ [∆u(x,y)−\gamma(x,y)u(x,y)]^2}$$ where the ...
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34 views

How to solve the fractional polynomial optimization problem?

The optimization problem has a fractional of polynomials as the objective function, with linear constraint. For example, $\min\limits_{x,y}\quad ...
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1answer
41 views

Gradient of a sum of indicators

Say I have a function $\mathbb R^n \rightarrow \mathbb R$: $$f(w_1,\ldots,w_n) = n^-\sum_{i\in I^-}w_ix_i$$ with fixed $x_i\in\mathbb R$ (data), $I^-$ the set of indexes with negative sum operands ...
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89 views

Optimize to Find the Mahalanobis Distance to Minimize the Term

I have an optimization problem defined as following: Assuming we have a data set $ { \left\{ \left( {x}_{i}, {y}_{i} \right) \right\}}_{i = 1}^{N} $ where $ {x}_{i} \in {\mathbb{R}}^{d} $ and $ ...
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3answers
101 views

How to find a pareto optimal solution in a smart way (3 variables)

$\max\left( { 3x }_{ 1 }+4{ x }_{ 2 }+2{ x }_{ 3 } \right) $ ${ x }^2_{ 1 }+{ x }^2_{ 2 }+{ x }^2_{ 3 }\le 1 $ ${ x }_{ i }\ge 0 $ I have to find a Pareto Optimal solution, but I can't solve this ...
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29 views

Closest line to point after non-linear map

I have a map on a vector space $M(\vec{r})$, defined as below. All components (vectors, matrices, everything) are reals in the unit range $[0,1]$. The map $M(\vec r)$ is defined as the sum of an ...
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29 views

Can this problem be a form of Nonlinear Programming?

How we can reformulate blew problem as a form of nonlinear programming problems? $$ \begin{array}{ll} & \min&\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n} &\\ & ...
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95 views

Derivation of back-propagation equation $\frac{\partial E(\theta)}{\partial W^k}=x*\delta h^k+\tilde{h}^k*\delta y$ for convolutional autoencoders

I was reading the following paper on convolution stacked auto-encoders and they had the following convolution neural network (for auto-encoders, notice I didn't write the offset term [to avoid ...
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165 views

Bilinear Constraint

I would like to formulate the following Optimization problem. My question is focused on the constraint. Given a "typical" objective function, e.g.: $$ \min c^T v $$ s.t. $$ 0 = a_1 v_1 - a_2 v_2 + ...
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2answers
30 views

Bivariate optimal density

Consider any feasible $p:[0,1]^2\to [0,1]$ that allows discontinuities and the problem $$\min_{p(.)} \int_0^1\int_0^1 p(x,y)^2 dF(x) dG(y)$$ s.t. $$\int_0^1 p(x,y)dG(y)=k\phantom{0} for \phantom{0} ...
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1answer
91 views

Finding the root for a multivariate function.

Background In a practical problem I need to find the solution to: $$f(\bar{x}) - \bar{p} = \bar{0}$$ where $ f : \mathbb{R}^2 \rightarrow \mathbb{R}^2 $. I don't know the exact expression for $f$ ...
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0answers
37 views

Mapping sphere surface to a vector space such that distances are preserved?

I have a unit radius sphere (say in 3D) centered in origin. Thus the shortest distance between two points on the sphere is the geo-desic. Is there a transformation (linear or non-linear) on the points ...
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166 views

Shannon Entropy Maximization with Constraints

I have got a cumulative distribution function $F_X(x)=Pr(X<=x)$. This distribution is described by 2 parameters $\alpha, \beta$. We define $F_k$ as follows: $\forall k<=n_k, ...
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1answer
32 views

Would like some help formulating an optimization problem

I have a function $f$ that takes $n \geq 1$ positive real-valued arguments $\mathbf{a} \in R^n_+$. This function is defined for all amounts of inputs (e.g. $f(1)$ and $f(3, \pi, 17)$ are both valid) ...
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1answer
57 views

invert S shape logistic curve fitting

I have a function F(x) = a + b / (1 + exp(-(cx + d))) and a small data set containing 5 sample: x = [10.0, 5.0, 2.0, 0.5, 0.25] and y = [23, 24, 25, 26, 27]. How can I find the suitable parameters, ...
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14 views

Duality gap analysis

I solved a non-linear non-convex optimization problem via dual decomposition optimization using sub-gradient method. (my main goal is to solve the problem in a distributed way). I solve the same ...
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49 views

Convergence results for block coordinate descent methods

I am trying to solve the problem minimize $f(x)$ subject to $x_1 \in C_1, x_2\in C_2, ... x_m\in C_m$ where $x_1, ..., x_m$ are block subvectors of $x$, and $C_i$ are each closed convex sets (not ...
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1answer
30 views

Maximization: KKT on unbounded region

Solve the following NLP: $$\left\{\begin{matrix} \min & -3x+y-z^2\\ s.t& g(x,y,z)=x+y+z \leq 0\\ & h(x,y,z)=-x+2y+z^2z=0 \end{matrix}\right.$$ My attempt Using kkt conditions, we ...
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36 views

non-linearity and non-convexity

I am taking a course on linear regression online and it talks about the sum of square difference cost function and one of the points it makes is that the cost function is always convex i.e. it has ...
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1answer
82 views

Matrix norm in the objective of an optimization problem

I am stuck with the following optimization problem from research. The optimization problem have the following objective function: $\|Q-H\|_\infty$. Here $Q$ is a PSD matrix and $H$ is a symmetric ...
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1answer
44 views

Eliminate cases before calculting all KKT conditions

I have the following non linear programming to solve: $$\left\{\begin{matrix} \min & (x-3)^2 + (y-2)^2 \\ s.t. & x^2 +y^2 \leq 5 \\ & x+y\leq 3 \\ & x \geq 0\\ & y\geq 0 ...
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36 views

Non-linear least squares and Bundle Adjustment

In METHODS FOR NON-LINEAR LEAST SQUARES PROBLEMS, 2nd Edition, April 2004 by K. Madsen, H.B. Nielsen, O. Tingleff on page 17 it states: Given a $f: R^n \mapsto R^m$ with $m \geq n$ We want ...
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1answer
47 views

If $H$ is positive definite and $s^Ty>0$, then $s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1$

Let $H\in\mathbb{R}^{n\times n}$ be symmetric and positive definite $s,y\in\mathbb{R}^n$ with $s^Ty>0$ How can we show, that $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...