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 ...

learn more… | top users | synonyms

2
votes
0answers
38 views

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 ...
2
votes
0answers
60 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 } ...
2
votes
0answers
51 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 ...
1
vote
1answer
45 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? $$\frac{1}{2} \sqrt{\frac{x}{l}+\frac{l}{x}+4 x^2-2}$$ ...
0
votes
0answers
25 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 ...
0
votes
1answer
23 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 ...
0
votes
0answers
35 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 ...
0
votes
0answers
45 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 ...
0
votes
4answers
96 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 = ...
1
vote
1answer
34 views

Removing a max function in the constraints

Can the following problem be transformed into a linear programming problem: Find $x_1,..,x_N$ which maximizes the objective function $$\sum_{i=1}^{N}x_{i}\sum_{j=1}^{n_{i}}c_{ij}$$ subject to the ...
0
votes
1answer
43 views

MINLP optimization with matlab reaching different solutions every run

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 ...
0
votes
1answer
60 views

matlab MINLP optimization with ga

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. There are additional constraints but ...
2
votes
0answers
21 views

discrete nonlinear convex optimization relaxation over a dense set

Be a discrete nonlinear convex optimization problem $P$ \begin{align} \underset{x\in \mathrm{C}^n}{\mathrm{min}} \ \ \ f(x) \\ Ax=b \\ c \leq x \leq d \end{align} $C$ is a dense in $F$. Is solving ...
1
vote
0answers
21 views

Optimal value of decision variable leads to inconsistency

$\epsilon$ is a random variable with support in $(0.8,0.95)$ and pdf $f(\epsilon)$. The following equation arises out of a business problem: $ENP=800*A*E(\epsilon)+ 9000 - ...
1
vote
0answers
35 views

Minimization using logarithmic barrier function

I'm thinking of the quadratic problem(QP) \begin{align} &\underset{x\in \mathrm{R}^n}{\mathrm{Minimize}}\ \ \ \frac{1}{2}x^\top{}Qx + f^\top{}x\\ &\mathrm{subject\ to}\ \ \ \ a_ix \leq b_i\ ...
1
vote
0answers
31 views

Point spectrum of a nonlinear operator on finite dimensional space

Given a nonlinear operator $T$ mapping $\mathbb R^n$ into itself, are there any known conditions on $T$ ensuring that the number of points in its point spectrum is upper bounded by the dimension $n$?
1
vote
0answers
25 views

Optimal time control for the system of two non-linear ODE

I have the following system of two non-linear ODE with one control variable (modified model of Lotka-Volterra): Here is $\alpha, \beta, \gamma, \delta$ - some constants, $u$ - control variable. ...
0
votes
1answer
84 views

A binary min-max optimization problem

I encountered a very special optimization problem for a practical application. We have a variable $$\mathbf{s}=(s_1,s_2,s_3, s_4)^T$$, where $s_i$ can only take $1$ or $-1$, and we also have a ...
0
votes
0answers
15 views

Polar cones' property [duplicate]

I am trying to prove: $A \subseteq B \implies B^\circ \subseteq A^\circ$ where $A^\circ$ is polar cone of $A$ ($A$ convex cone) and $B^\circ$ is polar cone of $B$ ($B$ convex cone)
1
vote
1answer
45 views

What does 'the level set is bounded' exactly want to tell?

'The level set is bounded.' occurs in many theorems and other places. I think I can understand the definition of 'level set' but I don't know what does 'it's bounded' want to tell me exactly in ...
1
vote
0answers
11 views

Finding a Sparse Binary Vector Which Satisfies a Vector Ordering

I have a research problem that I have reduced to the following problem which seems to be hard: Find $\mathbf{x}\in \mathbb{B}^N$ $(\mbox{where }\mathbb{B}= \{0,1\})$ which is $K$-sparse and which ...
1
vote
1answer
20 views

How to convert non-PSD matrix to PSD matrix?

I have a mixed-integer optimization problem with the following constraint matrix $Q_1$: \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 ...
1
vote
2answers
92 views

Local optimality of a KKT point.

Consider the problem \begin{equation} \min_x f(x)~~~{\rm s.t.}~~~ g_i(x)\leq 0,~~i=1,\dots,I, \end{equation} where $x$ is the optimization parameter vector, $f(x)$ is the objective function and ...
0
votes
2answers
58 views

How to get the perfect square for the following equation

The problem is defined as follows: $$ \min_X tr(X^T A X)-\alpha tr(X^T B) $$ I want to get the equal perfect square equation as that above, that is $$ \min_X \| X-C\|_F^2 $$ where $C$ is related to ...
3
votes
1answer
163 views

fantasy basketball model

i'm creating a fantasy basketball model (could be used in other games too) where we can project how well a player will do against another team even when the player hasn't played against a certain team ...
1
vote
1answer
54 views

Solving a Non-linear Multivariable System of equations

How would I go about solving a system of nonlinear equations where the highest degree is two? For example: $$f_1(x) = f_1(x_1, x_2,\dots, x_n) = 0,$$ $$f_2(x) = f_2(x_1, x_2,\dots, x_n) = 0,$$ ...
0
votes
1answer
40 views

Quadratic Program reformulation

I have the quadratic program $$\max\quad \mu^Tx+r_fx_0-\gamma \sum\limits_{i=1}^n |x_i-y_i|-\frac{\lambda}{2}x^TVx$$ $$\text{s.t. }\quad \mathbb{1}^Tx+x_0=1$$ where $\mu$, $r_f$, $\gamma$, $\lambda$, ...
1
vote
0answers
29 views

Newton's method: Is the change of parameter values between consecutive steps always decreasing?

Assume that I have a twice differentiable function $f(x)$ which I try to maximize with respect to $x$ (let's say $x$ is $k$-dimensional vector). When performing optimization via Newton algorithm, ...
1
vote
0answers
28 views

Are there any algorithms for simultaneous optimization of multiple objective functions?

I would like to minimize a set of similar objective functions $$f_\boldsymbol{s}(\boldsymbol{x}),$$ where $\boldsymbol{x} \in A \subseteq \mathbb{R}^M$ and the parameterization $\boldsymbol{s} \in S ...
0
votes
1answer
52 views

Roots of an equation using Maple

I am using Maple to find the roots of a non-linear equation in one variable. When I solve the equation, I get only 2 negative roots whereas if I plot the graph of the function, it also shows that the ...
2
votes
0answers
155 views

Direct multiple shooting (numerical optimal control)

please, Iam currently implementing direct multiple shooting method* and I need one simple but fundamental concept answered: When I want to provide not only objective funtion value (result of ODE ...
0
votes
0answers
52 views

SOCP or SDP optimization problem

I am studying an optimization problem \begin{equation} \mathbf{w}^* = \text{argmax} \sum_{d=1}^D \log \bigg( \frac{|\mathbf{f}_d^H\mathbf{w}|^2+c_1}{|\mathbf{f}_d^H\mathbf{w}|^2+c_2} \bigg)\\ \\ ...
3
votes
2answers
42 views

Sensitivity of polynomial global minimizers with respect to perturbations in the coefficients.

I'm trying to find the value of a global minimizers of a multivariate polynomial (4 variables) of high order numerically. The numerical values of the coefficients are coming from noisy measurements ...
0
votes
0answers
47 views

Newton's method for unconstrained optimization applied to a quartic function in R2

I am faced with the task of applying Newton's method to the following problem: $$ \text{min} ~~~~~ 8x_1x_2+\frac{1}{4}(x_1-x_2)^4 $$ where $x \in \mathbb{R}^2$. For clarification, the Newton method ...
2
votes
1answer
17 views

About constrained optimization

I've the following optimization problem:$$\min f(\theta_1,\theta_2)=\frac{a}{\cos\theta_1\cdot v_1}+\frac{b}{\cos\theta_2\cdot v_2}$$$$\operatorname{sub}\quad a\cdot\tan\theta_1+b\cdot\tan\theta_2=c$$ ...
2
votes
1answer
86 views

Why is the conjugate direction better than the negative of gradient, when minimizing a function

In gradient descent we minimize a function $f(\textbf{x})$, by using the update rule: $$\textbf{x}_{t+1} = \textbf{x}_t-\alpha\nabla f(\textbf{x}_t).$$ We also know, that at each iteration we have ...
1
vote
1answer
21 views

A weird optimization problem

I've the following optimization problem:$$\max f(R,z)=R^2(a+z)$$$$\operatorname{sub}\begin{cases}R^2+z^2=a^2\\0\le z \le a\end{cases}$$ Once solved it gives $z=a/3$, ... Consider now the ...
0
votes
0answers
49 views

Can a positive definite kernel produce a kernel matrix which has negative eigenvalues?

(1) I've read that a symmetric matrix is positive definite when its associated eigenvalues are all positive. I am learning SVM lately, and have come to know a $d$th-degree polynomial kernel ...
0
votes
1answer
75 views

What is the correct change of variables to yield convexity in this nonlinear optimization problem?

$$ \text{min. } x/y \\ \text{s.t. } 2\leq x \leq 3 \\ x^2+y/z\leq \sqrt{y} \\ x/y=z^2 \\ x,y,z\geq 0 $$ To transform this problem into a nonlinear convex optimization problem, both the objective ...
1
vote
1answer
46 views

Why test problems in convex optimization are mostly random?

Very often people who compare performance of different algorithms in convex optimization use randomly generated data. For instance, this often happens in compressed sensing and signal processing. Is ...
0
votes
0answers
48 views

Solve: tanh(x) = a*x + b - most efficient way

I work on DSP code, where some equations are of form: tanh(x) = a*x + b (tanh or other hyperbolic functions) Currently I use Newton-Raphson method. Is there a better/faster method of finding ...
0
votes
1answer
40 views

How to linearize this mixed-integer nonlinear constraint

Can someone please help me to linearize the following nonlinear/nonconvex constraint: $\sum\limits_{n=1}^Na_n\rm{log_2}(1+x_ny_n)\le M\delta$ Here $a_n \in\{0,1\}$, binary integer variable $0\le ...
0
votes
1answer
46 views

How to linearise this nonlinear constraint

I want to linearize or convexify this following constraint. Here $c_t$ is binary integer variables, $p_t$ are continuous variable which are bounded. $\gamma$ is a continuous variable. $h_t$ and $V$ ...
0
votes
0answers
30 views

What is a good optimization algorithm/tool for otimization on Partially Ordered set?

Actually I'm interested to minimize following kind of functions: $f: U \rightarrow V$ where: $U$ is a vector space and $V$ is a Ordered vector space, i mean Partially Ordered Vector space. ...
0
votes
0answers
33 views

Solving large non-linear polynomial equation system

I have a 2 order equation system of 7 unknowns. It is constructed as this: F1=0,F2=0,F3=0...F7=0 of which F1=f1*f2,F2=f3*f4... And f1=a1*p1+a2*p2+a3*p3+a4*p4+a5*p5+a6*p6+a7*p7 a1~a7 are known ...
0
votes
1answer
46 views

Non linear programming

Could you please help me in solving the problem posted below. A company uses a raw material to produce two types of products. When processed, each unit of raw material yields 2 units of product 1 and ...
0
votes
0answers
30 views

Linear independence of equality constraint gradients in constraint qualifications

I'm, trying to get an intuitive feel for the various constraint qualifications for KKT points. Most of them seem to rely on the linear independence of $\nabla g_i(x^*)$ where $g_i$ are the equality ...
1
vote
0answers
31 views

Reduce degree of a high degree unconstrained binary term to quadratic unconstrained binary term

I'm working on a optimization project, in this project I have to convert higher order unconstrained binary polynomial to quadratic unconstrained binary polynomial. Can anyone give me a hint of how to ...
1
vote
0answers
54 views

Mixed-Integer Linear Programing : get the maximum constant associated to a non null variable

Does anyone know a way get the maximum constant associated to a non null variable using Mixed-Integer Linear Programing ? I would like to get the variable $a$ in this description : $$ i = 1,\ldots,m ...
1
vote
0answers
98 views

Line search Armijo, Wolfe, Strong Wolfe and Goldstein.

What are the articles (References) who proposed the line search of Armijo, Wolfe, Strong Wolfe, and Goldstein? Articles precursors of unidirectional searches?