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Questions tagged [optimization]

This tag is intended for questions on methods for the (constrained or unconstrained) minimization or maximization of functions.

2
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1answer
1k views

scipy optimize fsolve or root

I have a function: delt=1 #trial def f(z): return ((1-2*z)*np.exp(-delt/z))/(((1-z)**(2+delt))*(z**(2-delt))) I also have a variable: ...
0
votes
1answer
3k views

Backtracking-Armijo Line Search Algorithm

EDIT based on comments below: I add the mathematical formulation of my problem below. I am trying to solve an equation of the form $$ \partial_t f(x,y,t)= (\partial^2_x +\partial^2_y) f(x,y,t) \equiv ...
1
vote
2answers
38 views

Optimisation of purely integer quantity with bound-constraints for a 1D expensive function whose analytical form is not available

I have a computationally expensive objective function, whose analytical form is not available. The only input argument to the objective function is an integer variable. The goal is to compute the ...
2
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0answers
76 views

Literature on numerical solving based on multiple meshes?

Consider solving a differential equation system for 1D, 2D, or 3D. It involves various input and output "field" variables, which, more often than not, correspond to various physical quantities ...
2
votes
1answer
118 views

Minimize the number of unique elements in a vector

I was wondering if there is a simple or known way to minimize the number of unique elements in a decision variable (vector). Note that I'm not asking for minimization of nonzero elements (rank ...
1
vote
2answers
141 views

Numerical optimization algorithm with approximated derivatives

Suppose I have a energy functional E depending on X, where X is a N-dimensional real value vector and N could be very large (~=2000). I assume that there exists (at least) a (local) minimizer for E. ...
0
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1answer
98 views

Obtaining extra output argument(s) from the objective function used by fsolve in MATLAB

I have a MATLAB code (see below) that employs 'fsolve' from the optimization toolbox for a root finding problem. The bottleneck is that, within the objective function calculation, there is a ...
2
votes
3answers
589 views

How to get multiple solutions for a optimization problem using any kind of software

I have a optimization problem in which the optimal objective value occurs at multiple point in the feasible space. If I run my problem in LINGO software then it gives me the optimal objective value at ...
3
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0answers
251 views

Optimisation of matrix exponential

I have a 7000x7000 sparse matrix (scipy), which I want to exponentiate. I've tried using scipy.sparse.linalg.expm, which works quite well for smaller matrices (takes a few seconds for a 1000x1000 ...
1
vote
1answer
26 views

constraint satisfaction via an LD solution

I'm going through the article in the following link lately and one point confuses me a lot. https://arxiv.org/pdf/1509.05001.pdf So, the goal of this paper is to solve the following constrained ...
1
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0answers
57 views

constrained quadratic binary problems and quantum adiabatic evolution

I'm going through an article with title "Solving constrained quadratic binary problems via quantum adiabatic evolution" (reference 1). And there are several points confusing me a lot. This article is ...
1
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0answers
13 views

Optimizing estimator of composed functions when function is known

Note: This was cross-posted from comp-sci, as I didn't know this community existed! I have a problem which I'm looking to see if there is literature on: Consider three types of actors, a Director, ...
1
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0answers
41 views

Unclear definition of objective function for MPM

While reading this paper on MPM (page 48), the definition of the objective function $E(\boldsymbol v_{\boldsymbol i})$ didn't make sense to me, although I completely understand the definition of (what ...
0
votes
1answer
126 views

lagrangian dual and linear programming

I'm going through an article lately and there is one point which is very confusing. So, we have the following original constrained binary quadratic problem as the following. The pre-assumption of ...
4
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0answers
72 views

linear relaxation of an optimization problem

I'm reading an article lately, and there is one point which confuses me. So, we have the following constrained binary quadratic problem. min $x^{T}Qx$ with the constraints that $Ax≤b$ and $x\in {0,...
4
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0answers
105 views

Stochastic gradient descent for large deterministic optimization problems

The Wikipedia page for SGD describes optimizing a function $f = \sum f_i(\theta;x_i)$ by successively approximating gradients from random subsets of the data, while most literature poses the problem ...
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0answers
132 views

First approximation to the TSP in a non-complete Graph

I'm trying to solve the Travelling Salesman Problem in a non-complete graph $(G,E)$ using genetic algorithms. My problem is that I can't find a good first approximation by the usual greedy algorithms,...
1
vote
1answer
96 views

GAMS Optimization

I am writing a GAMS program where I am interested in using the value of a variable as a condition inside an another equation. Let's say I have two equations with two variables, $g_1(t)$ and $g_2(t)$, ...
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0answers
122 views

How to test for convergence (smoothness of Pareto front) in DEAP

In the DEAP algorithms (see documentation here), I notice that we need to specify the the number of generations (NGEN). I was advised that convergence has been achieved if the Pareto curve is smooth. ...
5
votes
1answer
135 views

Optimization of non-smooth, non-convex, locally Lipschitz functions of type exp(-abs(x))

What would be the numerical method of choice to find minima in a non-smooth, non-convex, locally Lipschitz function $f: \mathbb{R}^n\rightarrow \mathbb{R}$. The function $f$ is mostly smooth but ...
1
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0answers
54 views

Comparing the solutions to a multi-objective optimization problem

Suppose I have a multi-objective optimization problem, and I wish to find solutions using two different methods/algorithms. The result of each algorithms is a Pareto front. Comparing two different ...
1
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0answers
72 views

How does this Constrained Minimization algorithm work?

I don't fully understand the subsection 3.2 Constrained minimization of this paper. In particular, I don't understand the first step "Register active set" and the definition of projection $P(x)$. ...
3
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0answers
93 views

Optimization based integration for MPM

I'm considering implementing (just for simplicity) the unconstrained implicit optimization based integration for Material Point Method as described in Chenfanfu Jiang's thesis on MPM (the minimization ...
0
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1answer
168 views

What penalty function produces optimization-based Gaussian smoothing?

I have just read yet another introduction to image processing, which describes Gaussian smoothing from a convolution perspective and least-squares smoothing from an optimization perspective. It would ...
1
vote
2answers
550 views

LU decomposition of large dense matrices

I wanted to generate LU decomposition of large size dense matrices ($N>10^7$), the LU decomposition I'm currently using is based on Adaptive Cross Approximation and is taking very long time to ...
1
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0answers
159 views

Variable elimination in linear programming

I have a linear program of the form $$\underset{P,\;g}{\text{Minimize}}\hspace{3mm}c^Tg$$ \begin{align} \hspace{17mm}\text{Subject to}\hspace{3mm}AP_{\cdot,j}&=\begin{bmatrix} -g\\ d \end{...
3
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0answers
90 views

A least square problem with a fixed mean constraint and a subspace constraint

Let $V_1,\ldots,V_n$ be $n$ vector subspaces of a Hilbert space, $y_i\in V_i$ for each $i$ and $\overline{x}$ be a fixed vector. I want to solve the optimization problem: \begin{equation*} \begin{...
4
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0answers
58 views

Relaxing a variable in MIP

I have this MIP optimization problem, with couple of binary variables; however when I relax one of the binary variables the optimal solution of the objective does not change. But the solving time ...
4
votes
0answers
1k views

Understanding MATLAB's `fmincg` optimization function

I'm researching numerical optimization. Recently I've come across a variant of a conjugate gradient method named fmincg. The function is written in MATLAB and is ...
2
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0answers
60 views

Global optimization with known distributions of some variables

I'm solving simple single-objective multidimensional global optimization problem using various stochastic algorithms like Monte-Carlo, GA and other evolutionary approaches. The task is formulated as ...
2
votes
2answers
157 views

Approach to handle a quadratic constraint xy <= z

I have non-linear constraints like $ x_1x_2\leq x_3 $ where $ x_1,x_2,x_3\geq 0 $. The objective is linear, and all other constraints are linear, too. I know that I can transform the product as $ ...
3
votes
0answers
87 views

Finite difference method for coupled PDEs: optimizing performance (time step, iterations per step)

I'm solving coupled PDEs using finite difference method: Incompressible Navier-Stokes and the divergence-free induction equation (Maxwell's equations) with non-uniform electrical conductivity. The ...
6
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0answers
104 views

How to optimally choose points for multivariable Hermite interpolation?

I have a multi-variate, continuous function $f$ from $R^n$ to $R$, which I can query for its output for any input. I would like to create interpolation polynomial for it. In one-dimensional case ...
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0answers
93 views

preconditioning LBFGS?

I want to minimize an energy of the form $$V_1(\mathbf{x}) + V_2(\mathbf{x})$$ where $V_1$ is much stiffer than $V_2$. When I try to use LBFGS, convergence is extremely slow, as the solution ...
4
votes
3answers
216 views

How to debug a constrained optimization algorithm?

I have implemented a saddle point optimization problem based on the algorithm by Prof. Nesterov, primal-dual[1]. Unfortunately, it doesn't work. It seems it is converging. But unfortunately, not to ...
8
votes
1answer
113 views

reformulating inverse problem as multi-objective optimization

I'm working on an inverse problem for my Ph.D. research, for which I'll write the objective functional as $J(\theta) = E(G(\theta) - u^o)$, where $\theta$ are the parameters, $G$ is the forward map ...
1
vote
1answer
18 views

Break a set of numbers to minimum groups with each having a common divider

I have a set of numbers, all positive, integers obviously, and what I want to do is divide them in the least possible groups with each group having a GCD bigger than 1. The biggest possible number is ...
4
votes
2answers
165 views

Optimization of known function with respect to two unknown function arguments

I have a data set, composed of points $(x_i, y_i)$ for $i=1,N$. I also have a known function $F$, which maps these points $x_i$ to $y_i$ as such $F(x_i, a(x_i),b(x_i)) = y_i$, where $a(x_i)$ and $b(...
0
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2answers
357 views

Optimization using Python/MATLAB

Given the following optimization problem: Given positive integers $n_1, n_2, \dots, n_m$ and real numbers $P_1, P_2, \dots P_m$. Besides these, given positive real numbers $a_{i,j,l}$ for $1 \leq i \...
1
vote
1answer
99 views

SOCP: Recovering primal from dual

Consider the following second-order cone program (SOCP): $$ \begin{array}{rl} \min_x & c^\top x\\ \mathrm{s.t.} & \|A_ix+b_i\|_2 \leq c_i^\top x+d_i \ \forall i \end{array} $$ Suppose I solve ...
3
votes
1answer
168 views

Minimizing linear objective on intersection of convex sets

Suppose I wish to solve the following optimization problem: $$ \begin{array}{rl} \min_{\mu\in\mathbb{R}^n} &\mu^\top c\\ \textrm{subject to} & \mu\in C_1\cap C_2\cap\cdots\cap C_k, \end{array} ...
2
votes
2answers
90 views

Question concerning accumulation point

Suppose that we have to find an optimal solution $x^*$ to an optimization problem involving some function $f$, such that $0\in\partial f(x^*)$ where $\partial$ denotes the subdifferential. Let $(x_n)...
3
votes
1answer
132 views

Why are Hamiltonian dynamics used in MCMC?

In Hamiltonian Monte Carlo, Hamiltonian dynamics are used to generate new proposals from the current state. I understand why these dynamics are used as opposed to random walk behavior to generate ...
0
votes
1answer
276 views

Nonlinear integer program with linear constraints

I'm trying to perform inference over a subset of the latent variables of a hierarchical hidden Markov model I built. I've derived the relevant optimization problem, but it's a pretty nasty piece of ...
0
votes
1answer
123 views

Optimization with Yalmip [closed]

I would like to solve in Matlab the following optimization problem $$\begin{array}{ll} \text{maximize} & \bigg\| \displaystyle\sum_{l=1}^{2}\alpha_l \int_{\tau_{m+l-1}}^{\alpha_1\tau_m+\alpha_2\...
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0answers
68 views

vectorizing optimization or root finding [closed]

I need to find the roots of a function. I am currently using scipy.optimize.fsolve ...
5
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0answers
398 views

Precision of ratio constraint in a linear program [closed]

I am trying to solve a LP in which one of my constraints is of the form $$\frac{A(x)}{B(x)} = 1$$ I transform this constraint into a linear one $$A(x) - B(x) = 0$$ However when CPLEX solves the ...
1
vote
1answer
28 views

Parameter identification for regression model

Consider the following regression model : Y = AX + BU where the size of Y is $N \times n$, A is $N \times n$, X is $n \times n$, B is $N \times n$ and U is $n \times 1$. The matrices X,Y and U are ...
4
votes
3answers
2k views

Beating typical BLAS libraries matrix multiplication performance

A dull matrix multiplication algorithm where we use the formula $$C_{ij}=\sum_{k}A_{ik}B_{kj}$$ By literally following this in 3 loops we'll get a very slow program, because we don't utilize ...
3
votes
2answers
297 views

optimization problem. Monte Carlo stochastic method or another one?

I have the following problem, there is an objective function f() depending on 7 variables x=(x1,x2,...x7), so f(x)=f((x1,x2,...x7)) and I want to find the combination of variables that minimize the ...