Questions tagged [constrained-optimization]

Questions about optimization problems subject to additional constraints.

73 questions with no upvoted or accepted answers
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6
votes
0answers
168 views

An invertible matrix that minimizes the norm of the product with a given matrix

Given a fat matrix $B \in \mathbb{C}^{n \times m}$ (where $m > n$) with full row rank, I would like to find (numerically) a full-rank matrix $A$ that minimizes the Frobenius norm of the product $A ...
5
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0answers
77 views

Solution of constrained system of ODEs

Can someone point me in a direction to solve this kind of integral constrained system of ODEs. \begin{align} &\int_0^{1/2}\dot{y}^2(t)=p\\ &2\lambda_1\ddot{y}(t)+\pi cos(\pi y(t))=0\\ &y(...
5
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0answers
354 views

Optimization on the manifold of stochastic matrices

So I have an optimization problem of the form $$\text{maximize}\hspace{3mm}f(A):{\bf R}^{K\times K}\rightarrow{\bf R}$$ $$\text{subject to}\hspace{19mm}A^T{\bf 1}=\bf{1}$$ $$\hspace{33mm}A\geq 0$$ ...
5
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0answers
514 views

Best way to add a positivity constraint to Newton's Method

So given an objective function $f({\bf x})$, I would like to include a positivity constraint when I perform the fixed point iteration: $${\bf x}^{(t+1)}={\bf x}^{(t)} - \text{H}_f^{-1}\nabla f({\bf x}^...
5
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0answers
77 views

numerical solver for stochastic optimal control problems

can any one recommend numerical solver (c/c++ library preferred) for stochastic optimal control problems? For deterministic optimal control I found something like that: http://abs-5.me.washington.edu/...
4
votes
1answer
99 views

Non-negative least squares with very small numbers

(I have asked this question on StackOverflow previously but it has been pointed to me that CSSE or MSE could be more appropriate) I have to solve a constrained optimization problem of the following ...
4
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0answers
70 views

Fast approximate solver for vehicle routing problem

I need to solve capacitated asymmetrical vehicle routing problem with time windows on ~30k points. Time limits for calculations are 2 hours. I've tried using Clarke and Wright savings algorithm, it is ...
4
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0answers
55 views

Obtainting KKT for QSDP for the trace inequality constraint

I am working on developing my own solver(for implementation on hardware), based on IPM for following problem: \begin{equation} \begin{split} \min_{X} \; \frac{1}{2}&\|X\|_F^2 + trace(CX)\\ \text{...
4
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0answers
164 views

How to apply an integrated constrain condition in FEM?

I'm running some simulation using FEM. In my model I need to apply a constraint condition to the governing equation. My governing equation similar to the diffusion equation as below: $$\frac{\...
4
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0answers
160 views

Eigenvalue-style optimization with quadratic constraints

Suppose $A\in\mathbb{R}^{n\times n}$ is symmetric and positive definite and that we have several symmetric matrices $B_i\in\mathbb{R}^{n\times n}$ that are low-rank and indefinite. I need an ...
3
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1answer
84 views

Solve two-player game - minimize the l-infinity norm of a matrix-vector product

I have a matrix $M$ with non-negative real entries, and I would like to minimize the objective function $$\Phi(v) = \|Mv\|_\infty,$$ where $v$ is constrained to be a probability vector, i.e., $v_1+\...
3
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0answers
47 views

Least-squares fit of explicit parabolic sheet to data points

For a given set of data points $$\{(x_i, y_i, z_i)\}$$ there exists some $$f_{ABC}(x,y)=Ax^2+Bxy+Cy^2$$ that minimizes $$\sum_i(f_{ABC}(x_i,y_i)-z_i)^2$$ $A$, $B$, and $C$ can be found quickly ...
3
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1answer
63 views

Plotting optimum as a function of parameter in the objective

I am trying to minimize a 2d function using scipy.optimize. Specifically I want to plot the minimum value of the function fun as a function of the parameter wjk. The problem is that I cannot pass wjk ...
3
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0answers
129 views

Automatically generate constraints for trajectory optimization

This is a follow up to my previous post here I'm interested in performing trajectory optimization from the problem mentioned in abov link. I want to supply the following as dynamical constraints to ...
3
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0answers
28 views

How to set up and solve acceleration-limited trajectory optimization problems?

I've been trying to learn how to solve simple acceleration-limited trajectory planning problems. I'm working in C++ and I've been using the Eigen library to do linear systems solving. I'm doing the ...
3
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0answers
22 views

Sequence planning with 3 machines

together! First of all, I have to mention that because of my background as an Industrial Engineer, I have limited abilities in mathematics, but am disciplined enough to expand myself from ...
3
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0answers
48 views

Constraint solver vs Bayesian optimizer for fast discontinuous processes

I have a complex domain-specific process that accepts inputs: 10-500 inputs, where each input is of type: enum: choice between multiple string or numeric values int: integers float: floating point ...
3
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0answers
85 views

Solve ODE with non-negative and maximization constraints

My task is to solve $$\eta_k\frac{d^2C_k}{dz}(z)=-e_k, k = 1,2,3$$ $$C_k\ge0$$ $$C_1(0)=0, C_2(0)=A, C_3(0)=0$$ $$C_1(L)=B, \frac{dC_2}{dz}(L)=0, \frac{dC_3}{dz}(L)=0$$ with $$e_1 = -\beta_1-\beta_3$$ ...
3
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0answers
95 views

Minimize number of rectangles that cover all the points

I have a 2d distribution of moving points with known trajectories represented in a 640x480 image. Here is the initial state: I have to find the minimum number of rectangles with fixed dimensions (...
3
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0answers
201 views

Optimal Control using Dynamic Programming - Optimizing for Furthest Distance

So I have been investigating a problem to get a glider with control of its elevator to fly as far as possible from any given initial state. To keep this simple, we will view this in 2D space with the ...
3
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0answers
75 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,...
3
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0answers
102 views

Applications of algorithm for solving systems of equations with uncertainty

We have been developing algorithms for detecting "robust" zeros of multidimensional functions $f: X\to\Bbb R^n$ where $X$ is an $m$-dimensional domain in $\Bbb R^m$. More precisely, for a given $f$, ...
3
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0answers
175 views

Find constrained vectors maximizing angles between them - methods?

This is related to a question I had asked earlier, with the distinction that earlier I did not have a non-linear objective functional to minimize. The problem is reproduced below with added ...
3
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0answers
73 views

Why not use this simpler variant of Stepwise Regression?

In stepwise regression, you step predictor by predictor, each time selecting the one with the greatest correlation with the measurement, subtracting greedily to leave a residual with no correlation to ...
2
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0answers
37 views

Lexicographically order matrix into a vector

I am trying to implement the algorithm contained in this article here. It is about solving a 2 and 2.5D Fredholm integral, focused on bidimensional NMR experiments. I've made significant progress, ...
2
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0answers
43 views

Biconvex problem whose objective function depends on only one variable

I am solving the following biconvex problem: $$\min_{x,y} f(y)$$ $$s.t. ~~ g(x) \leq 0$$ $$~~~~~h(x,y) = 0$$ $$x \in X, y \in Y$$ where $X$ and $Y$ are compact convex sets, $g(x)$ and $f(y)$ are ...
2
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0answers
53 views

How to find two points within defined region in this constrained optimization problem?

I am doing a project related to robotics where I am using fmincon function from matlab to minimize the distance between the points ...
2
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0answers
42 views

Sensitivity Lagrangian solution general case

I have asked this question already on maths and mathoverflow. Just a question about a literature reference. I am writing a paper for engineers. Usually for the Lagrange multiplier problem $$ \...
2
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0answers
279 views

Convergence of a very large non-linear least squares optimization

(note: I also posted this question on stackoverflow before finding this community here, which seems a better place for it) I'm trying to solve the following problem: I have a lot (~80000) surface ...
2
votes
0answers
61 views

How to model pedestrian flow through subway systems?

I'm a New Yorker and take the subways every day. I have a growing interest in understanding the distribution of paths people take on the subways to work every day. I.e. if there are $n$ subway ...
2
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0answers
95 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{...
2
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0answers
209 views

Using Line Search Method for Constrained Optimization

Suppose we have a $f(x)$ to be minimized (we only know that $f(x)$ is three-differentiable), and a feasible, convex set of $S$ such that all $x$ belong to $S$. Using line search method, how we can "...
2
votes
0answers
42 views

$(max(0, f(x)))^2$ or $(max(0, exp(f(x))))^2$ for soft constraints with Gauss-Newton

I need some kind of inequality constraint in my optimization problems (rude version of SVM for example or skeleton based mesh fitting). However hard constraints is not suitable for me because ...
2
votes
0answers
68 views

Convergence of KKT equations for discrete parameter estimation problems

Consider a discrete constrained optimization problem: $$ \mathbf{q}_*^h= \arg \min {\cal J}^h(\mathbf{x}^h[\mathbf{q}^h],\mathbf{q}^h) $$ subject to the (weak-form) constraint $$ F^h[\mathbf{x}^h(\...
2
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0answers
388 views

Solving constrained BVP, singular Jacobian

The boundary value problem is $$ \begin{cases} \dot{x}_i = \begin{cases} (0.5D^{-1}\psi)_i, \text{ if }(0.5D^{-1}\psi)_i \le 0 \\ 0 \text{, otherwise} \end{cases} \\ \dot{\psi} = 2\Sigma x \\ x(0) =...
2
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0answers
92 views

non convex, non linear optimization involving matrix differential equation solution

I'm trying to develop an inferential procedure for a multivariate dependent Markov process. Basically, the procedure could be considered as a non linear regression, with a known dependence structure ...
2
votes
0answers
121 views

Hessian eigenvalues in 4D-VAR data assimilation

I am using variational data assimilation (4D-VAR) to estimate emissions of anthropogenic greenhouse gases using a rather complex atmospheric transport model. Hence, the optimal solution to my problem ...
2
votes
0answers
50 views

Inverted value is not consistent with expectation

We have a group of observations $$y = f(x_1, x_2, x_3) \enspace .$$ We have also a forward model $y = f(x_1, x_2)$. The forward model does not include $x_3$ because $x_3$ might include dozens of ...
2
votes
0answers
69 views

Fortran solver for the Sparse LSE problem

I was wondering if there is a Fortran library that contains a solver for the Sparse LSE(linear equality-constrained least squares) problem $$ min_{x}\|Cx-d\|^2 \text{ subject to } Ax=b $$ where $A$ ...
2
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0answers
81 views

About Convex Geometry

A consistency notion in constraint programming: Let $P = (X, D, C)$ be a CSP. Given a set of variables $Y \subseteq X$ with $|Y| = k -1$, a locally consistent instantiation $I$ on $Y$ is $k$-...
2
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0answers
131 views

Inverse problem with a rank-1 update

I hope you can help me out with this. I have to find the solution x to an inverse system $$ x=A^{-1}b $$ This inverse problem is basically a least square problem with a rank-1 update. $$ x=[uv^{T}...
1
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0answers
22 views

Bipartite Euclidean Matching simple to implement approximate algorithm

I am looking for a simple to implement algorithm for the bipartite euclidean matching problem (or an implementation of any practical algorithm). I am aware of Agarwal's paper, but I would like to ...
1
vote
0answers
44 views

How to speed up the Mixed-Integer Quadratic Program process?

Currently, I am solving a problem in the format: M is an integer as well. The problem that troubles me is that X is a vector in {0,1} with a size of 7000. I use the solver in https://github.com/...
1
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0answers
32 views

What is the generalization of the resource allocation problem I'm dealing with here?

I'm dealing with a problem as follows: I have a finite set of money 𝑚 to spend over 𝑟 different raffles, and I need to spend approximately to my budget, with the goal of maximizing my probability ...
1
vote
0answers
43 views

Optimization of centers and radii of circles under the non-collision constraint

I want to optimize a function w.r.t. $n$ circles parametrized by centers and radii: $$\min_{C, R} f(C,R)$$ where $C\in\mathbb R^{n \times 2}$ and $R\in\mathbb R^n$. For example, ...
1
vote
0answers
69 views

Ramp least squares estimation

With some given $s$ value, let \begin{equation} \begin{aligned} h(\beta)&=\min(\sum_{i=1}^n(Y_i - X_i\beta)^2, s)\\ &=\sum_{i=1}^n(Y_i - X_i\beta)^2-\max(0, \sum_{i=1}^n(Y_i - X_i\beta)...
1
vote
0answers
16 views

Long AMPL model preparation time

We deal with a large-scale linear optimization problem (~50000 variables and ~4000000 constraints). We use AMPL Studio modeling environment for problem modeling and then calling linear solver (CPLEX, ...
1
vote
0answers
36 views

Minimizing a polynomial with millions of monomials

I need to minimize a single polynomial $P(x_1,x_2,...,x_n)$ with the constraint that for each $i$, $0\leq x_i \leq 1$. The number of variables in my practical problem is at most $50$. The degree is at ...
1
vote
0answers
34 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 ...
1
vote
0answers
38 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))...