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.
3,001
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Convexity of function, Hessian
Im trying to understand convexity of a given function $$f(x)=x_1^2+x_2^2+3x_1x_2+10x_1-11x_2+5.$$
My initial thought was to only take the second derivatives and check that $f_{xx} \geq 0$, and $f_{yy} ...
- 593
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30
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Are second-order conditions in optimization really needed?
I’m participating in optimization course and a lot of time is spent proving second-order conditions for unconstrained and constrained problems. To me these conditions feel rather unnecessary since I ...
- 339
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25
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Linearize or approximate a square root constraint with binary variable [closed]
I am working with a Mixed Integer Nonlinear Model, and to use CPLEX for solving it, I must linearize or approximate the following non-linear constraint.
$S_{i,j,s}$ is binary variable and $D_{j,s}$ is ...
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How can I convert this non-convex optimization problem into a convex one? [closed]
I am dealing with a non-convex optimization problem, getting stuck at turning the objective function into a concave function w.r.t. x. (a, b, c, d, and e are all real numbers.)
\begin{align}
\max \...
- 9
1
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2
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An optimization problem and proof of necessity
The problem is formulated as follows
$
\max f\left( x_k,y_k \right) =\log _2\left( 1+Ax_1y_{1}^{2} \right) +\log _2\left( 1+Ax_2y_{2}^{2} \right) -\log_2 \left( 1+ACD^2 \right)
\\
s.t\,\,x_1+x_2=C,
\...
- 21
1
vote
1
answer
36
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How to reformulate negative power of posynomial as a geometric programming constraint?
Currently, I am working as a network optimization engineer. From what I see, most of my optimization task can be fulfil effectively by using Geometric Programming (GP). However, I am running into a ...
0
votes
2
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107
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Solve $ y'' + \lambda y^2 = 0$
Solve $ y'' + \lambda y^2 = 0$
Attempt 1 : if $\lambda =0 $ .then it's trivial to solve.
If $ \lambda <0 $ ,then $y '' \ge 0$
In particular when $y '' > 0 $ for some interval .
Let $ y''= e^{u(...
- 2,795
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35
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Bound on the min of the sum of cosines at two rationally related frequencies
I am interested in the minima of the sum of two frequencies:
\begin{equation}
\Delta = \min_t\left[\cos(t) + \delta \cos\left(\frac{p}{q}t+\phi\right)\right]
\end{equation}
$\phi, \delta\in\mathbb{R}$,...
- 167
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46
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Maximisation of a function involving the ratio of quadratic and linear forms
I have a function to maximise
$\boldsymbol{x}'A\boldsymbol{a} - \frac{\boldsymbol{x}'B\boldsymbol{x}}{\boldsymbol{x}'\boldsymbol{b}} - \frac{\boldsymbol{x}'C\boldsymbol{x}}{\boldsymbol{x}'\boldsymbol{...
- 73
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Using Block Coordinate Descent for Convex Optimization with Quadratic Constraints
In solving the optimization problem $$min _{X, Y} f(X, Y) $$ $$\text { s.t. } \operatorname{tr}\left(X X^T\right)+\operatorname{tr}\left(Y Y^T\right) \leq P$$
, where both $f_X(Y)$
and $f_Y(X)$
are ...
- 13
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29
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Nonlinear KKT Optimization Problem
Check whether the 1st-order necessary conditions for
optimality hold at the optimum point (1, 0) of the following NLP:
$$min \; f(x) := -x_1$$
$$s.t. \;g_1(x) := -(1-x_1)^3 + x_2 \leq 0$$
$$g_2(x) := -...
- 152
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1
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Bilinear Constraints with Spectral Norm
Consider the following set of constraints
$$
\rho_{k+1} \leq \rho_k \|\tilde{A}_k\| + \varepsilon_k \|D\|, \quad \forall k = 0,\ldots, N-1,
$$
where $\varepsilon_k > 0$ is a constant, $D\in\mathbb{...
- 314
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Image of a system of nonlinear inequalities under a linear map
Suppose we have a nonlinear feasible set $F$ in $\mathbb R^8$ given by
all $(x_{11},x_{12},x_{21},x_{22}, y_{11},y_{12},y_{21},y_{22})$
satisfying
$x_{ij},y_{ij}\geq 0$ and
$x_{ij}^2+y_{ij}^2\leq 1$, ...
- 131
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1
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A single objective optimization problem, but involves higher order terms of undetermined parameters due to the existence of recursive equations
The following problem of solving undetermined parameters troubles me a lot, and I do not know where to start, hope someone could give me some hints,
Given a sample data $(p_0,q_0,p_M,q_M)$, the ...
- 11
1
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0
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58
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Proving the triangle inequality from a given fact
We're given fact (1):
$$\forall x,y \in \mathbb{R}^n, \sqrt{1+|x+y|^2} \leq \frac{\sqrt{1+|2x|^2} + \sqrt{1+|2y|^2}}{2} $$
Using the Jensen's Inequality and fact (1), please prove the triangle ...
- 33
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How to quantify the stability of an optimization to compare different sets of basis functions?
I'm minimizing some value stochastically (Monte Carlo methods) using different sets of basis functions to fit some unknown function needed to calculate this value. The dimensionality of the parameter ...
- 1
1
vote
1
answer
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Quickly solving a system of integral equations using numerical methods/ analytical way to speed up
I am not sure if this is more suitable for math stackexchange or stackoverflow but here we go: I have a system of integral equations with unknowns $\theta_1, \cdots, \theta_5$:
$$\int_{-\infty}^{\...
- 1,769
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Sequential projection onto circumferences
Let $S = (s_1, \dots, s_n) \subset \mathbb{R}^{n \times m}$ be a sequence of euclidean spheres,
where
$s_k = \{ x : \| c_k - x \|^2_2 \leqslant r_k^2\} \subset \mathbb{R}^m$ is the sphere $s_k \in S$ ...
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31
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Adaptive Runge Kutta for gradient descent optimization
A version of this question was asked a couple of years ago here, but I am still not clear on why this is not used more widely (or seemingly at all).
Problem statement: Suppose you have some ...
- 1
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1
answer
61
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Distances between specific hyperplanes as nonlinear optimization problem
For any sets $X, Y \subseteq \mathbb{R}^{n}$, the distance between them is defined as $$d(X,Y) = \operatorname{min}_{\mathbf{x} \in X, \mathbf{y} \in Y} = \sqrt{(\mathbf{x} - \mathbf{y})^{T}(\mathbf{x}...
- 545
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62
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An optimal control problem coupled with target allocation
Motivation: What we are doing is to let a group of unmanned surface vessels (USVs) return the fixed berths at the dockside. To make USVs intelligent and us convenient, we want to let them pick their ...
- 11
0
votes
2
answers
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Nonconvex quadratic objective with quadratic constraint
I need some hints to prove whether problems P1 and P2 are equivalent or not. Any help will be appreciated.
$
P1: \min_{x} | x^H C^H C x - d |^2 ~~ {s.t.} ~~ x^H P^H P x = a,
$
$
P2: \min_{x} | x^H ...
- 778
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33
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Non-convex programming
I want to solve a non-convex optimization problem of the form :
\begin{array}{cl}
\displaystyle \min_{x} & f(x)\\
\textrm{s.t.} & c(x) = 0,\\
\end{array}
where $f$ is a concave smooth function ...
- 11
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answers
45
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Number of iterations needed for the method of steepest descent
The function $f(x,y) = 4x^2 + 2y^2 + 2xy -4x + 6y$ has a unique global minimizer at $(x,y) = (1, -2)$
Starting at $(5,2)$ how many iterations of the steepest descent method would it take, at least, to ...
2
votes
1
answer
43
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Low-dimension matrix approximation?
I have a matrix $\mathbf E_1 \in \mathbb R^{m\times n_1}$. I would like to approximate it with a matrix $\mathbf E_2 \in \mathbb R^{m\times n_2}$ with $n_2 < n_1$, where "approximate" ...
- 389
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29
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Minimizing a Function with Nonlinear Constraints
I am trying to minimize the function $f(x, y) = \frac{1}{x} + \frac{1}{y}$ subject to the constraints:
$$
x^3 + \frac{x}{x + y} y^3 - c \leq 0,
$$
$$
x \geq 0,
$$
$$
y \geq 0.
$$
I have attempted to ...
- 125
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1
answer
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Minimizing a Function with Nonlinear Constraints
I am trying to minimize the function $f(x, y) = \frac{1}{x} + \frac{1}{y}$ subject to the constraints:
$$
x + \frac{x}{x + y} y - c \leq 0,
$$
$$
x \geq 0,
$$
$$
y \geq 0.
$$
I have attempted to use ...
- 125
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Optimizing a Complex Project-Employee Assignment Function (Pure nonlinear 0-1 programming)
I'm working on optimizing a project-employee assignment problem involving a complex objective function. I'm seeking help to understand the best approach to maximizing this function.
The objective ...
- 1
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How can I estimate unknown parameters in a linear model?
In the paper "Linear Pushbroom Cameras" by Gupta and Hartley they introduce a linear parametric model (equation 12):
$$
K(V_x,V_y,V_z,f,p_v,\alpha,\beta,\gamma) =
\begin{bmatrix}
1/V_x &...
- 865
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27
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Linear optimization objective with little-o notation in constraint
I am trying to solve a problem related to matrix decomposition. In essence, it is a maximin problem of the following form
$$ \max_x \min_{j} \left[f_{j=1}^{(n)}(x),f_{j=2}^{(n)}(x),...\right]$$
where ...
- 272
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answer
48
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Unimodality of multivariable quadratic functions
Are multivariable quadratic functions necessarily unimodal, so local optimum is always the global optimum? When the Hessian is indefinite then can we conclude the quadratic function is either ...
- 95
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How does this optimal x change with this parameter?
Consider an optimization problem
$$\max_x V = f(x,a) + g(x,a)$$
where $\frac{\partial^2 f}{\partial x^2},\frac{\partial^2 g}{\partial x^2} < 0$
Let $x^*$ (the optimal $x$) be such that
$$\frac{\...
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1
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0
answers
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Converting $x^3$ Optimization to an Equivalent LP Problem
Suppose we want to find the minimum of a a strictly increasing function $f: [-a, a] \to \mathbb{R} $, for some $a > 0$, which is also concave in $[-a, 0]$ and convex in $[0, a]$ (exactly like the $...
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optimization based on complex matrix differentiation
I'm calculating the gradient of a loss function with respect to some complex matrices. The loss function is defined as $\mathcal{L}=||I(x)-\widetilde{I}(x)||_2^2$, where $I(x)$ is the output from a ...
- 11
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Show that $x_{k}$ is the solution of the augmented Lagrangian problem.
Consider the problem of minimizing $f(x)$ such that $h(x)=0$.
Let $x_{k}$ be the global minimum of the aumented Lagrangian problem for certain $\lambda_{k}$ and $\rho_{k}$ and consider that $h(x_{k})=...
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answers
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How to determine which eigenvectors of the Lagrangian Hessian are constraints?
I am trying to implement a constrained optimisation algorithm where the constraints are not satisfied at the beginning. I am using a Lagrangian multiplier method:
$$L(x, \lambda) = F(x) - \sum \lambda ...
- 267
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25
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Concave maximization over $d$-dimensional simplex.
Can either an analytic solution or the dual be characterized for the following concave maximization:
$v:= \underset{w \in \Delta_d}{\max} \sum^{d}_{i=1}\frac{1}{\sqrt{1+b_i/w_i}}$
where $\Delta_d$ ...
0
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0
answers
20
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SOCP and polynomial time constraints
Given a multi-variable function $f$ to minimize subject to some constraints, I am confused after reading few papers and the Wikipedia entry about SOCP(second order cone programming).
I already know ...
user411780
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0
answers
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How to set up a convex concave procedure for the minimization of $abc$?
From this post, it seems that there are a lot of advantage of approximating nonconvex problem with the convex concave procedure. Out of curiosity, suppose that I have a simple problem that is
$\begin{...
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1
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Computational Difficulty with Solving System of Equations in an Optimization Problem with 3 constraints (2 active & 1 inactive)
I was given the following problem:
Minimize: $$f = -2x + 3y^2$$
Subject to: $$g_1 = (x-1)^2 + y^2 > 1 \\g_2=(x-1)^2 + y^2 \leq 4\\g_3 =x \geq 0 $$
Currently, I am trying to find a minimizer ...
- 33
0
votes
2
answers
89
views
Optimizing a function using Python
I have the following optimization problem where ( a ) refers to an index in an array of length ( K = 4 ):
$$
\sup_{w \in W} \inf_{\lambda \in L} \sum_{a=1}^{K} w_a. \frac{(\mu_{a} - \lambda_{a})^2}{2}...
- 21
0
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16
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Constructing smooth paths between points in codomain
I have a differentiable non-bijective mapping f from $x \in R^T$ to $y = f(x) \in R^T$.
Say I have some positions $x_s^*$ and $x_e^*$ with corresponding mappings $y_s*$ and $y_e^*$.
Questions:
I ...
- 743
3
votes
3
answers
263
views
Finding a bound for optimal solution of a quadratic optimization problem
Suppose that $A\succeq 0,$ $0\le y$ and $\|y\|\le \|x_\rho\|, \mu \in \mathbb R^n,$ and $e$ be the vector of ones. My goal is to show $\{x_\rho\}$ is bounded when large enough $\rho$ goes to $+\infty$,...
- 366
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votes
0
answers
15
views
Distributed newton optimization algorithm for m x n dimension
newton optimization algorithm to find the local minimum x∗ of a non-linear function f(x) with iteration sequence of x0→x1→x2...→x∗ all ∇2f(x_{k}).
considering the x has m, and n with an index of i ...
2
votes
0
answers
22
views
Maximizing a semi-concave function
A function $f:\mathbb{R}^d\to\mathbb{R}$ is called semi-concave if $x\rightarrow f(x)-\frac{\lambda}{2}\|x\|^2$ is concave for some $\lambda>0$.
Suppose I want to maximize a function $f$ which is ...
- 299
2
votes
1
answer
71
views
What is so interesting about the Armijo-Rule or the Wolfe-Conditions for choosing the right step size?
Right now I am taking a course on nonlinear optimization where we currently talk about step size rules(Armijo-Rule and Wolfe-Conditions).
I also had a course 1 year ago about statistical machine ...
- 455
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votes
0
answers
63
views
What is so interesting about the lagrange-newton method?
I have taken a course on nonlinear optimization. There we discussed penalty and barrier methods for nonlinear optimization. In the end my lecturer gave a short conceptual idea of SQP by introducing ...
- 455
-1
votes
1
answer
68
views
Non-linear parameter optimization using Python
I have a model that generates the curve represented by the red squares the data represented by the black circles.
The model curve (red squares) depends on some parameters to fitting. Is there any ...
- 99
5
votes
1
answer
239
views
What kind of interesting properties that make exponential cone attractive?
I am a network engineer who is studying some optimization problems in the field of communication theory mostly for pleasure. Out of pure curiosity, I see that there is some optimization problem in ...
1
vote
1
answer
40
views
Proposition 3.2. of Bertsekas' paper about Lagrange multipliers
This is a problem about the proposition 3.2 of Bertsekas' paper below.
Bertsekas D P, Ozdaglar A E. Pseudonormality and a Lagrange multiplier theory for constrained optimization[J]. Journal of ...
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