Questions tagged [optimization]
This tag is intended for questions on methods for the (constrained or unconstrained) minimization or maximization of functions.
0
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0answers
13 views
Possible application of Polya's urn on real data (for Portfolio optimization)
I wanted to find some more information of this topic, but I found very little.
I might be interested in optimizing a stock investment portfolio. Maybe I could use beta or some other common risk ...
1
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0answers
21 views
Examples of problems that cannot be formulated as optimization problems
An optimization problem has 3 main components: decision variables, constraints and an objective function. Such a problem can be mathematically modelled and solved using an optimization solver. For ...
0
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1answer
50 views
Solve multi-dimensional optimization problem using basinhopping
I am searching for an optimization solution, which is a 8d vector representing 4 complex elements, where each element is within the complex circle with maximal radius 1.2.
The objective function is:
...
3
votes
2answers
73 views
Optimization techniques for expensive multi-variable functions
I'm working with a finite element model in which I'm interested to minimize the average temperature at a surface. I have 15 independent variables in my model, including geometry, materials, flows, ...
1
vote
3answers
115 views
Find a solution of large system of inequalities
I have a large system of homogenous inequalities involving 33 real unknowns of the form
$$
\vec{F}(z_i)^T \cdot \vec{X}>0\,
$$
where $\vec{X} = \left(x_1,...,x_{24}\right)^T$ are the unknowns and ...
2
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0answers
42 views
Non-parametric models as solutions to Partial Differential Equations
In the realm of scientific computing, there are a plethora of techniques developed to solve Partial Differential Equations (PDEs). Many of the popular methods are variants of common techniques such as ...
4
votes
1answer
59 views
What is the name for this type of constraint?
I have what would be a straightforward mixed-integer linear programming problem, except for the fact that some of the constraints are of the form $f(x_1,x_2,x_3,\ldots,x_n) < c$, where $f$ is 'take ...
0
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0answers
12 views
Cost functions to judge time/memory/accuracy tradeoffs
I am working on an interesting algorithm: Its absolute error is exponential in a parameter $j \in \mathbb{N}$, and for a given $j$, I have complete freedom to choose between an $\mathcal{O}(1)$ time-...
0
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0answers
90 views
How to solve extremely large scale linear system
I have a extremely large scale liner system equation. It stems from the minimization of
$$
E(\{\widetilde{C_{i}}\}) = \sum_{i}\left[ \left( 1-\psi _{i}\right) \left\| \widetilde{C_{i}}-\sum_{j\in{N_{i}...
0
votes
1answer
16 views
Minimize squared error of linear function
Let $M$ be a $m \times n$ matrix, $x$ a $n$-vector, $y$ a $m$-vector, and $\|\cdot\|_2$ represent the $L_2$ norm (i.e., Euclidean norm). Given $M,y$, the goal is to find $x$ that minimizes the ...
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0answers
27 views
Minimizing the ratio of two specific non negative quadratic convex functions
$F$ is $m\times m$ diagonal, with real non negative elements
$D$ is $n \times m$ complex
$P$ is $n \times 1$ complex
$A$ is $m \times 1$ complex.
Minimize $\Gamma(A)$, with respect to $A$.
$$\...
2
votes
1answer
82 views
Is there an optimization scheme/algorithm that converges, for this non-convex scenario but with some special properties
I have a smooth function $f(x) = \frac{g(x)}{h(x)}$ that is the ratio of two smooth convex functions $g(x)$ and $h(x)$. It is known that $f(x)$ has a global minimum, achieved at the unique point $x_0$....
1
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0answers
45 views
Simplification of an optimization objective
Let $G(V,E)$ is a weighted simple graph, where $V$ and $E$ are the set of vertices and Edges. The graph is undirected. Let $A \in \{0,1\}^{n\times n}$ and $W \in R_+^{n\times n}$ be the adjacency ...
0
votes
1answer
32 views
Vehicle Route assignment with capacity constraint
Problem Background
I'm trying to find a solution/model to the following problem:
Let's consider a cellular network (mobile network, ie., hexagonal cells) denoted $N$ composed of $|N|$ cells. Each ...
7
votes
1answer
183 views
Lack of quadratic convergence in Newton's method
It is well-known that Newton's method can converge quadratically, if initial guess is close enough and if the arising linear systems are solved accurately.
I am applying Newton's method to highly ill-...
3
votes
1answer
71 views
Minimize a function with sparse Hessian
The problem I am trying to solve involves minimising a function with respect to a large number (probably 10,000+) of parameters. I can cheaply compute both its Jacobian and its Hessian. The Hessian is ...
0
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0answers
32 views
In-exact line search
In my class notes, the author says:
"If $f:\mathbb{R}^n \to \mathbb{R}$ is bounded below and $p_k$ is a descent direction and the $\alpha-\beta$ also known as Armijo-Goldstein condition is met then ...
0
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0answers
30 views
Is this a form of stochastic gradient descent?
I want to minimize the following with respect to parameters $B$.
$$\sum_{k = 1}^{K} f(A_{k}, B)$$
where $A_k$ are $K$ different data-sets and $B$ is a matrix of parameters.
Can I do this by a ...
0
votes
1answer
23 views
Gradient ascent method with a constant step size?
I'm trying to use the gradient ascent method on a convex function like the multivariate-Normal density function with respect to its parameters (the original is a bit more complicated), something ...
-3
votes
1answer
56 views
2
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2answers
183 views
Best software to do big number calculations quickly
I am trying to do some work on some math conjecture. I am testing the conjecture numbers using very large math numbers (100+ digits ). I am currently using python to test these numbers.
In the ...
2
votes
1answer
47 views
Question about strange outputs from the CVXPY solver
I am familiarizing myself with CVXPY, and encountered a strange problem. I have the following simple toy optimization problem:
...
3
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0answers
58 views
Given a list of intervals, find region that is contained by the largest number of those intervals
Start with 1d case. Say I have lots of 1d intervals $[s_i, e_i]$ and I want to find an interval $[s^*, e^*]$ to maximise the count of interval $i$ such that $[s_i, e_i]\supseteq [s^*, e^*]$.
1d case ...
3
votes
0answers
71 views
Nonlinear least squares and regularization
Consider the nonlinear least-squares minimization of a vector of $n$ residuals $\mathbf{f}$ in $p$ parameters $\mathbf{x}$:
$$
\min_{\mathbf{x}} || \mathbf{f}(\mathbf{x}) ||^2
$$
This can be done with ...
0
votes
1answer
64 views
Is this a knapsack problem?
I have a set of $K$ keywords. Each of this keywords can have set of bids from $1\$,\dots,N\$$. For each bid for a keyword, it will get a specific amount of clicks and a specific cost. Clicks and Cost ...
0
votes
1answer
65 views
Minimize cost with Levenberg-Marquart method
I want to minimize a cost function of the form,
$$
\min_{q,t}\left(q^T\left(\mathcal A + \mathcal B\right)q + t^T\mathcal C t+\delta t+\varepsilon Q(q)^TW(q)t+\lambda\left(1-q^Tq\right)^2\right)
$$
...
0
votes
1answer
60 views
How to derive the optimal bayesian solution to a model of two normal distributed populations
In the "Introduction" section of the paper Support-Vector Networks, it mentioned Fisher's solution to a model of two normal distributed populations:
My questions are:
How to derive equation (1)? I ...
1
vote
1answer
59 views
sum of absolute difference constraint in optimization problem
I am writing a model for an optimization problem. I need to write the following constraint:
$$\sum^{N - 1}_i \lvert (a_i - a_{i+1}) \rvert \leq 2\, .$$
How to write this constraint (or linearize)?
...
1
vote
2answers
114 views
Method to find PDE equation coefficient satisfying mean solution?
What is the best approach to go about solving a PDE problem of the type
\begin{equation}
k^3\Delta u - k(\mathbf{1}\cdot\nabla u) = 0\, ,\\
u=g\; \text{on}\; \Gamma_D\, ,\\
mean(u) = u_\text{...
1
vote
1answer
50 views
Nature of stationary points of a Lagrangian fuction
I would like to extremize a certain function $f$ with respect to a parameter $x$, under constraints $g_1(x) = 0, ..., g_m(x)=0$. In order to achieve this, I consider the Lagrangian function $L(x, \...
-1
votes
1answer
68 views
Compute affine transformation between two sets of points
Consider two sets of points $P = (P_1, ...,P_n), \ Q = (Q_1, ..., Q_m) $ included in $\mathbb{R}^3$.
I'm looking to compute an optimal affine transformation that "maps" $Q$ to $P$, although the sets ...
3
votes
1answer
70 views
Nonlinear least squares when some parameters are linear
Consider the least squares problem,
$$
\min_{\mathbf{a},\mathbf{b}} || \mathbf{f}(\mathbf{a},\mathbf{b})||^2
$$
where $\mathbf{a},\mathbf{b}$ represent the unknown parameters to be found. In my ...
1
vote
0answers
351 views
Good C++ optimization library for BFGS
To implement maximum likelihood estimators, I am looking for a good C++ optimization library that plays nicely with Eigen's matrix objects. Eigen has some capability of interfacing of its own but if ...
2
votes
1answer
47 views
Techniques to remove a function from Levenberg-Marquardt when it is against box constraints
I have a somewhat large (20+ dimensional) root finding problem that I'm solving with Levenberg-Marquardt. One of the functions has box constraints on [0, 2]. When it is against those bounds it will ...
2
votes
0answers
25 views
Where can I find sample data for large linear programming optimization problems?
I am doing a comparison of different algebraic modeling langues (AMPL, AIMMS, GAMS, Pyomo) in both theoretical and practical terms. As a practical experiment I am trying to measure problem model ...
1
vote
1answer
82 views
Why does Newton's method with Linear Equality Constraints use KKT condition?
Goal: Optimize convex function $f(\vec{x})$ subjected to constraint $A\vec{x} = \vec{b}$ starting at a point $\vec{x}_0$ that satisfies the constraint.
The problem only has equality constraint. Why ...
1
vote
0answers
46 views
Order of a principal term
In Yurii Nesterov's Introductory Lectures on Convex Optimization, there is a bound for the total number of iterations for some process. See page 109:
$$\left[\frac{1}{\ln(2(1-\kappa))} \ln\frac{t_0-t^...
1
vote
0answers
32 views
Logging vs outputs in iterative optimisation
I'm coding an iterative algorithm of constrained continuous optimisation. An augmented Lagrangian algorithm (outer) calls a bound-constrained L-BFGS-B algorithm (inner), which calls a line search ...
3
votes
0answers
48 views
Nonlinear functional optimization in radial coordinates
I am currently implementing classical density functional theory for a radially symmetric system.
In mathematical terms, I am searching for a function $f(r)$ that minimizes a functional $\Omega[f]$.
...
2
votes
0answers
76 views
How to numerically optimize affine transformations?
I need to optimize affine transformations for of a set of triangles using energy function based on the connectivity.
The energy of an edge $e_j$ between triangles $T_a, T_b$ is given by
$$
E_j = \...
-1
votes
1answer
26 views
R function or package for carrying out maximum likelihood techniques in random effect models
I am applying optim() function in R to obtain maximum likelihood estimates of the fixed effects and random effects in a model with bivariate random effects. The ...
3
votes
0answers
104 views
Difference between Dishonest Newton method and Very Dishonest Newton method
What is the difference between the Dishonest Newton method and the Very Dishonest Newton method? Is there a difference or do they mean the same thing?
I have tried searching for this on the internet ...
1
vote
0answers
33 views
Efficient numerical optimization of an “almost separable” function
I have come across an optimization problem with the following objective function:
$$f(x_0,y_0,z_0,x_1,y_1,z_1,...,x_N,y_N,z_N) = \sum_{i=0}^N f_i(x_i,y_i,z_i, \alpha(x_{i+1}-x_i) + \beta(y_{i+1}-y_i))...
2
votes
1answer
61 views
Convergence rate and complexity for convex minimization problem
In Yurii Nesterov's Introductory Lectures on Convex Optimization, there is a description of the rate of convergence and corresponding upper bound for the analytical complexity of a minimization ...
2
votes
1answer
32 views
Log-transformation of decision variables in parameter estimation
I am trying to find the diffusion coefficient ($D$) and the partition coefficient ($KLP$) using experimental data of desorption of a pollutant from a film into a liquid. This process can be modelled, ...
3
votes
0answers
32 views
Stochastic conjugate directions to improve convergence in narrow valleys
My question concerns a specific statement in this paper:
N. N. Schraudolph and T. Graepel, "Conjugate Directions for Stochastic Gradient Descent," in Int. Conf. Artificial Neural Networks, Berlin, ...
4
votes
1answer
388 views
How does fmincon in MATLAB calculate gradients?
I am trying to solve numerically a constrained optimisation problem in MATLAB, and I am wondering how the fmincon function calculates gradients when one isn't ...
1
vote
1answer
38 views
What is a good way to select a small subset (say 50) of items from a large pool of items (say 5 million) while minimizing an objective function?
I have 5 million items that have 10 features (all continuous and not categorical) each and would like to select a small subset of these items. Ideally, I want to manually specify 10 features of my own ...
-1
votes
1answer
81 views
Simulation-based Optimization vs PDE-constrained Optimization
What is the difference between Simulation-based Optimization and PDE-constrained Optimization?
Would studying a text on Simulation-based optimization be sufficient to understand and apply both?
3
votes
1answer
67 views
What is a “good enough” method of assigning values to n variables subject to basic bounding constraints while maintaining relative weights?
Given triples of $n$ floating point values
$$(\min_1, \max_1, w_1), \dots, (\min_n, \max_n, w_n)$$
and a value $V$, what is a good algorithhm to assign values $v_i$ to each of the triples such that ...
