Questions tagged [constrained-optimization]

Questions about optimization problems subject to additional constraints.

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12 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 ...
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25 views

Implementation method selection for sparse constrained linear least squares or quadratic programming

I need to slove one optimization problem of quadratic programming. The number of optimization variables is about 16,000. The constraints include equality constraints and inequality constraints. I have ...
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1answer
35 views

Constraint programming problem with conditional constraints and some unknown indicator variables

I have an interesting little problem that I believe can be formulated in terms of optimization or constraint programming. I have a few dozen variables $a$, $b$, $c$ ... and a set of constraints that ...
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1answer
35 views

Why is quadratic penalty method used for equaltiy-constrained optimization?

When one equality-constrained optimization is formulated, the method of Lagrange multiplier will be the choice for me. In Chapter 17 from the book Numerical Optimization, quadratic penalty method can ...
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1answer
230 views

Can this simple quadratic optimization problem be turned into a simple eigenvalue problem?

I'm interested in a type of problem on this form $$\min_{x} x^{T}Ax+x^{T}b \quad \text{s.t} \quad x^{T}x=1 $$ where $A$ is positive definite. As you can see, if it weren't for the $x^{T}b$ term in the ...
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39 views

Balanced constraint

In several optimization problems, we found what we called the balanced constraint(c^T • x = z ) For exp: C^T • 1 = 0 (C is a binary Matrix) Can I have an intuitive explanation about the concept and ...
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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/...
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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 ...
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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+\...
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1answer
57 views

Linear system with an l1-norm constraint

I have a saddle-point system of the form \begin{equation} \begin{bmatrix} A & B \\ B^T & O \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} f \\ \vec{0} \end{bmatrix}, \end{...
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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 ...
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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 ...
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21 views

Python: Getting second output variable from minimizing a computationally intensive function on first outputs

I have a function in python that is quite computationally expensive to evaluate, of the form: ...
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2answers
99 views

Using MILP to place a set of primers along a genome

Define variables $p_i,u_i\in\{0,1\}^G$, for $i=1,\ldots,8$ and $G=30000$. Let $v$ be a constant vector also in $\{0,1\}^G$, with approximately 25% of its entries equal to $1$ (randomly located). Let ...
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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 ...
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2answers
73 views

How to minimize $(x-a)^2+(y-b)^2$ subject to $ \sqrt{a}+\sqrt{b}=\sqrt{2}$?

I am not sure if this is on-topic here, but I am trying. Let $x,y$ be positive real numbers. I am trying to find $$ \min_{\sqrt{a}+\sqrt{b}=\sqrt{2}}(x-a)^2+(y-b)^2$$ I tried using Mathematica for ...
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1answer
34 views

Avalability of SNOPT optimization solver

I'd like to know if SNOPT solver is available free of cost for academic research in any of the optimization software packages. I came across a few softwares that have SNOPT, but those require a ...
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1answer
102 views

What's the right choice of variable settings for setting up my optimal control problem?

This is a followup to my previous question here I have the following dynamical system, $\frac{d \phi}{dt} = -M^TDM\phi \tag{1}\label{1}$ $\frac{d \hat\phi}{dt} = -M^T\tilde{D}M\hat \phi \tag{2} \...
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138 views

Setting up optimization problem in GEKKO

I have the following dynamical system, $\frac{d \phi}{dt} = -M^TDM\phi \tag{1}\label{1}$ $\frac{d \hat\phi}{dt} = -M^T\tilde{D}M\hat \phi \tag{2} \label{2}$ $\eqref{1}$ represents the exact ...
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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, ...
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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 ...
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2answers
107 views

Solving a parameter estimation problem using trajectory optimization

This is a follow-up to my previous question here I've the following system of equations for studying information flow in the below graph, $$ \frac{d \phi}{dt} = -M^TDM\phi + \text{noise ...
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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, ...
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1answer
176 views

Underdetermined Minimum Volume Enclosing Ellipsoid

Given three vectors in $\mathbb{R}^{512}$, my task is to compute a Minimum Volume Enclosing Ellipsoid (MVEE). I have tried Kachiyan's algorithm, but it requires at least as many vectors as there are ...
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1answer
66 views

Optimization with the constraint of rank=1

I have the following matrix $$ A = [x_1, x_2, ..., x_n], $$ where $x_i \in \mathbb R^n$. But I know the relationship that \begin{align} x_2 = s_2 x_1 \\ x_3 = s_3 x_3 \\ ... \end{align} where $s_i$...
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1answer
58 views

Could the convex problem be tackled by CVX?

I want to solve the convex optimization as follows: \begin{align} \underset{X_1,X_2}{\min} &\ -\frac{1}{N}\sum_{i=1}^N\log\det\left(I+H_i^HX_2H_i\right)-\log\left[1+h^H(X_1+X_2)h\right]\\ &\...
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1answer
120 views

Gradient descent in constrained optimization of barrier function

This question may be too basic, but I was wondering if it is possible to implement simple methods such as gradient descent or its variations to find the minimum of barrier functions in constrained ...
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2answers
280 views

How to solve calculus of variations problems numerically?

For example, how to solve the well-known isoperimetric problem (i.e., to enclose the largest area with a fixed-length curve)? We can simplify things a bit and fix the two ends of the curve at $[a,0]$,...
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1answer
62 views

Analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^T A z$, where $A=uu^T+vv^T$

Let $u$ and $v$ be column vectors of size $n \gg 1$ (not both zero), and consider the matrix $A:=uu^T+vv^T$ Question What is an analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^TAz=\arg\max_{\|z\...
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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)...
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3answers
180 views

Maximize a function of an orthogonal matrix

I'm trying to write up a small code that, given a set of normal vibrational modes for a molecule, will convert them to localized vibrational modes. To do this I'm following the procedure from J. Chem. ...
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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, ...
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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 ...
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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 ...
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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 ...
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1answer
342 views

Simultaneously maximize and minimize

I am virtually new to optimization (saw it years ago in a very shallow course) and now I came across a problem that I believe would require from it. The problem is I don't know exactly how to proceed. ...
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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 ...
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42 views

Optimization (best input variables search) for a non-smooth non-linear unknown function

I am trying to optimize a system that monitors and advises a user multiple times over a certain period of time depending on changing outside factors. The systems behavior can be altered by 5 ...
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1answer
145 views

How to solve a 4th order nonnegative LASSO problem?

I need to solve the following 4th order nonnegative LASSO problem: $$ \min_{x \geq 0} \quad || |Ax|^2 - b ||^2 + \lambda ||x||_1 $$ where $|\cdot|^2$ denotes element-wise squared. $A$ is small size (e....
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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 ...
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1answer
80 views

How to add extra constraints to a linear system for probabilities?

Background: I have an equation which looks like as follows: $W \times P = R$ $$\left[\begin{array} &{1}&{0}&{0}&-\frac{w_{1}}{w_{o1}} &\dots &{0} &-\frac{w_{1}}{w_{0} } \...
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238 views

Solving a non-convex optimization problem using fmincon

I am trying to solve a non-convex optimization problem using fmincon(). At each iteration, I am iteratively looking for the optimum value and when the termination ...
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1answer
197 views

Why would BFGS converge to a local minima of a non-convex function but maintain a large gradient?

I'm using BFGS to optimize a smooth but non-convex function $f$ that is computed by a simulation. The simulation also gives me a semi-analytical gradient $g$, which is verified by the numerical ...
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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 $$e_1 = -\beta_1-\beta_3$$ ...
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1answer
291 views

How do I solve the matrix equality constrained optimization problem using Lagrangian multipliers?

Solve the following minimization problem in $\mathbf{X} \in \mathbb{R}^{m \times n}$ $$\begin{array}{ll} \text{minimize} & \frac 12 \| \mathbf{X}\mathbf{X}^T -\mathbf{A} \|^2_\mathcal{F}\\ \text{...
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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 ...
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1answer
53 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 ...
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1answer
57 views

Research articles on MultiObjective Non-Linear Programming (MONLP)

I'm looking for papers dealing with multi-objective non-linear programming which could help me implement an algorithm to solve my problem. My problem is : Maximize $f(x) = c \cdot x$, while ...
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1answer
69 views

Formulate and solve a simple conic programs in cvxpy language [closed]

Let $r,\epsilon > 0$ and $a, b \in \mathbb R^n$ with $\|a\|_2 \le r$. Define $C(a) := \{x \in \mathbb R^p | \|x+a\|_2 \le r,\;\|x\|_\infty \le \epsilon\}$, and assume it is non-empty. Question (A)...
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105 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: ...

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