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Questions tagged [nonlinear-optimization]

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 trying to solve particular problems.

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. Usually, non-linear optimization problems are much harder to solve than linear ones.

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How do I derivate a function in the Norm?

I have a function $$f(x) = \frac12\|G(x)\|_2^2$$ where $G(x): \mathbb R^n \rightarrow \mathbb R^n$ is a twice continuous differentiable function. I want to determine $\nabla f(x)$ in terms of $G(x)$ and $G'(x)$. How do I derivate that norm?
Pablo Elias
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Nonlinear conjugate gradient descent for maximization

I have a multivariable function that I have defined the cost function and its gradient with respect to the variable vector. Let's call the cost function $f(\overrightarrow{x})$, and variable vector $\overrightarrow{x}$. I have been using nonlinear…
strahd
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Quadratic Programming Problem

Given the quadratic programming problem \begin{align*} min \quad 0.5(x_1-a)^2+0.5(x_2-b)^2+0.5(x_3-c)^2\\ s.t \quad 4x_1+5x_2=20\\ 8x_1+3x_2+12x_3=24 \end{align*} Use the method of projection to obtain analytically the optimal solution in terms of…
Andre Jackson
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Multivariable constrained optimisation

How to determine if a solution exists for a multivariable (3 variables ) optimisation problem with 2 constraints? What conditions should i check?
Aishwarya Iyer
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Is there something like "Integer non-linear programming" without mixed-integers

I just wanted to know whether problems exist that belong to the category "Integer non-linear programming" without belonging to the category "Mixed-integer non-linear programming"? While having ecoutered several "Integer linear programming" problems…
PeterBe
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Maximization to minimization with constraints

I have a maximization problem which is given below: \begin{alignat}{2} &\!\max_{x,y} &\qquad& {x+2y}\\ &\text{subject to} & & {x^2+y^2} = 1\\ & & & \end{alignat} And I need to convert it into a minimization…
Yukti Kathuria
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qualification constraints and Slater's condition

I'm considering the following minimization problem: $$ \begin{cases} min f(x) \\ s.t.\ \ \ \ g_{i}(x)\le0 & x \in R^n, i \in I={1,..m} & f,g_{i} \in C^1 \end{cases} \\ $$ Given $x \in S $ where S is the feasible region $ Set: \\ D1(\bar x)=\{d…
Giacomo
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Minimize $z=2x_1+3x_2-x_1^2-2x_2^2$ subject to $x_1+3x_2\le 6$, $5x_1+2x_2\le 10$, and $x_1,x_2\ge 0$

How can we minimize $z=2x_1+3x_2-x_1^2-2x_2^2$ subject to $x_1+3x_2\le 6$, $5x_1+2x_2\le 10$, and $x_1,x_2\ge 0$? I need to know the steps to solve or at least the guidelines as I am really new to nonlinear equations and I am trying this on my own.
Gopi
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minimizing a non-linear function when the constraint is not differentiable

Minimize the distance from $(x_0, y_0)$ in the 4th quadrant to the curve $y = |x|$ I know that the minimization problem is: minimize $$f(x,y)= (x-x_0)^2 + (y-y_0)^2$$ subject to$$ g(x,y) = y - |x|= 0 $$ Since the constraint g(x,y) is not…
user319373
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Difference between Hessian and approximation

In literature about trust region methods I found the following statement: First they define for $t\in (0,1)$ $f(x_k+p)=f(x_k)+\nabla f(x_k)^Tp+\frac{1}{2}p^T\nabla^2f(x_k+tp)p$ and by using $B_k$ as an approximation to the Hessian (with $B_k$…
sqlman
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Could someone help me with this Lagrangian linear programming problem?

I am struggeling with this problem: Power Generation Company PowGen owns three generation units, each of them with the following cost function Cost function Consider the basic CED without losses (Ploss =0) and without generation limits. a. What are…
Maxioloco
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If f is coercive and $\mathrm{epi}(f)$ closed, $\min f(x)$ has at least one solution

Let $\mathrm{epi}(f)=\{(x,\alpha)\in \mathbb{R}^{n+1}| f(x)\le\alpha\}$ Prove or disprove: If $f$ is coercive and $\mathrm{epi}(f)$ closed, the problem $\min f(x), x\in \mathbb{R}^n$ has at least one solution. If $f$ is coercive, then $\lim_{\|x\|…
user396039
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MINLP alternatives

I have a problem in which I need to allocate an amount of a service and each allocation consumes time. The mathematical formulation would be something like: $$ min: f=\sum_s^S{\sum_h^H{Amount_{s, h} \cdot Used_{s, h} \cdot…
Santi Peñate-Vera
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f is continuous if and only if epi(f) is closed

Let $epi(f)=\{(x,\alpha)\in R^{n+1}| f(x)\le\alpha\}$ Prove or disprove: f is continuous if and only if $epi(f)$ is closed. Can someone give me a hint?
user396039
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2 answers

Optimize $\sum a_k x_k^2$ under the constraint $\sum x_k =c$, with $a_k > 0$

Optimization, under costraint $f(x_1,\ldots,x_n)=\sum a_k x_k^2$ under the constraint $\sum x_k =c$ with $a_k > 0,$ $k=1,\ldots,n$ The associated Lagrangian is : $L(x_1,\ldots, x_n)=f(x_1,\ldots,x_n)+ \lambda(\sum x_k -c)$ How to deduce that:…
Paul-henri Guet
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