Questions tagged [constrained-optimization]
Questions about optimization problems subject to additional constraints.
3
votes
0answers
32 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 ...
1
vote
1answer
51 views
How to add extra constraints to a linear system for probabilities？
Background：
I have an equation which looks like as follows:
$W \times P = R$
$$\left[\begin{array} &{1}&{0}&{0}&-\frac{w_{1}}{w_{o1}} &\dots &{0} &-\frac{w_{1}}{w_{0} } \...
0
votes
0answers
45 views
Solving a non-convex optimization problem using fmincon
I am trying to solve a non-convex optimization problem using fmincon().
At each iteration, I am iteratively looking for the optimum value and when the termination ...
2
votes
1answer
69 views
Why would BFGS converge to a local minima of a non-convex function but maintain a large gradient?
I'm using BFGS to optimize a smooth but non-convex function $f$ that is computed by a simulation. The simulation also gives me a semi-analytical gradient $g$, which is verified by the numerical ...
0
votes
0answers
38 views
Augmented Lagrangian Techniques Frank-Wolfe Algorithm [python]
I'm trying to solve the convex quadratic problem (quadratic min cost flow problem) using the Frank-Wolfe algorithm.
$$ \min\{x^TQx+qx: Ex=b,\quad 0\leq x \leq u\} $$
The standard algorithm is okay ...
3
votes
0answers
53 views
Solve ODE with non-negative and maximization constraints
My task is to solve $$\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$...
1
vote
1answer
123 views
How do I solve the matrix equality constrained optimization problem using Lagrangian multipliers?
Solve the following minimization problem in $\mathbf{X} \in \mathbb{R}^{m \times n}$
$$\begin{array}{ll} \text{minimize} & \frac 12 \| \mathbf{X}\mathbf{X}^T -\mathbf{A} \|^2_\mathcal{F}\\ \text{...
1
vote
0answers
36 views
Biconvex problem whose objective function depends on only one variable
I am solving the following biconvex problem:
$$\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 ...
0
votes
1answer
36 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 ...
1
vote
1answer
26 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 ...
0
votes
1answer
43 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)...
2
votes
1answer
51 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:
...
2
votes
0answers
50 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 ...
4
votes
0answers
65 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(...
0
votes
1answer
36 views
Can LINCS algorithm be used for colliding molecules?
Supposing that one molecule is static and one is dynamic,
can the dynamic one be solved with LINCS for its shape (angle, bond length) constraints and also keep collisions with static molecule off, ...
0
votes
1answer
79 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 ...
3
votes
1answer
39 views
Improve optimization speed for a set of similar problems: Quadratic programming with a warm start
I am repeatedly solving quadratic program,
$x^T Q x$ with time dependent linear constraints $Ax=b_t$.
Dimension of $x$ is around 10000 and there are around 50 constraints. I want to solve the ...
1
vote
1answer
68 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\, .$$
How to write this constraint (or linearize)?
...
1
vote
1answer
60 views
Nature of stationary points of a Lagrangian fuction
I would like to extremize a certain function $f$ with respect to a parameter $x$, under constraints $g_1(x) = 0, ..., g_m(x)=0$. In order to achieve this, I consider the Lagrangian function $L(x, \...
2
votes
1answer
48 views
Techniques to remove a function from Levenberg-Marquardt when it is against box constraints
I have a somewhat large (20+ dimensional) root finding problem that I'm solving with Levenberg-Marquardt. One of the functions has box constraints on [0, 2]. When it is against those bounds it will ...
1
vote
1answer
99 views
Why does Newton's method with Linear Equality Constraints use KKT condition?
Goal: Optimize convex function $f(\vec{x})$ subjected to constraint $A\vec{x} = \vec{b}$ starting at a point $\vec{x}_0$ that satisfies the constraint.
The problem only has equality constraint. Why ...
1
vote
0answers
33 views
Logging vs outputs in iterative optimisation
I'm coding an iterative algorithm of constrained continuous optimisation. An augmented Lagrangian algorithm (outer) calls a bound-constrained L-BFGS-B algorithm (inner), which calls a line search ...
2
votes
0answers
33 views
Sensitivity Lagrangian solution general case
I have asked this question already on maths and mathoverflow.
Just a question about a literature reference. I am writing a paper for engineers.
Usually for the Lagrange multiplier problem
$$
\...
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:
$$f(x_0,y_0,z_0,x_1,y_1,z_1,...,x_N,y_N,z_N) = \sum_{i=0}^N f_i(x_i,y_i,z_i, \alpha(x_{i+1}-x_i) + \beta(y_{i+1}-y_i))...
6
votes
0answers
121 views
An invertible matrix that minimizes the norm of the product with a given matrix
Given a fat matrix $B \in \mathbb{C}^{n \times m}$ (where $m > n$) with full row rank, I would like to find (numerically) a full-rank matrix $A$ that minimizes the Frobenius norm of the product $A ...
4
votes
1answer
542 views
How does fmincon in MATLAB calculate gradients?
I am trying to solve numerically a constrained optimisation problem in MATLAB, and I am wondering how the fmincon function calculates gradients when one isn't ...
1
vote
1answer
119 views
Piecewise-Linear Quadratic Optimization for an “Almost Convex” Problem
I have a 7-14 dimensional piecewise linear cost function I'd like to minimize with two quadratic terms of the form:
$$
f(X) = X^tCX + d \sum_i |x_i-x^*_i|^2 + \sum_i P_i(x_i-x^0_i)
$$
$$ \sum_i x_i ...
-1
votes
1answer
85 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?
2
votes
0answers
225 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 ...
5
votes
1answer
171 views
Solvers for Quadratically Constrained Quadratic Programs (QCQP) with complex variables
I'd like to know whether there are any publicly available tools for solving QCQP with complex variables (and constraints therefore expressed through Hermitian matrices). What I have found so far is ...
1
vote
1answer
189 views
How to use CSDP to express a semidefinite program?
I am trying to use CSDP and am struggling with it. Consider, for example, the following semidefinite program
$$\begin{array}{ll} \text{minimize} & 0\\ \text{subject to} & Q - A' Q A - \...
1
vote
1answer
346 views
FMINCON Step Size Tolerance
I get following error after implementing the attached code.
Error Message
"fmincon stopped because the size of the current step is less than
the default value of the step size tolerance but ...
1
vote
0answers
42 views
Space covering optimization
I have the following problem:
In the space $E=\{1, 2, \dots, N_x\} \times \{1, 2, \dots, N_y\}$, I want to define $N_R$ rectangles $R_k=\{x_k^0, \dots, x_k^1\}\times\{y_k^0, \dots, y_k^1\}$ which ...
3
votes
2answers
146 views
How do I check if a loss function can achieve its minimum?
For example, the convex function $f(t)=e^{-t}$ doesn't achieve its minimum 0 on the real line. In a linear regression with $p$ predictors $X$, the loss function $f(\beta)=||Y-X\beta||^2$ achieves its ...
2
votes
1answer
28 views
Constraints 'exactly/at most one non-zero element' without binary variables
In a much larger MINLP problem, I have set of variables $\{a_{ij}\}_{m,n}$, such that $0 \leq a_{ij} \leq 1 $ for all $i$, $j$, which I could think of as a matrix, for which I have two requirements:
...
6
votes
1answer
220 views
What is required of the objective function in order to use Gauss Newton method?
From what I understand, the Gauss-Newton method is used to find a search direction, then the step size, etc., can be determined by some other method.
In addition to that, are the following ...
1
vote
1answer
572 views
How to Solve Optimisation Problems using Penalty Functions in Python
I am working on a implementing a simple quadratic optimisation problem:
$$\min _x \; {\underline{x}}^T Q {\underline{x}}$$
$$s.t. \,\quad {\underline{\mu}}^T{\underline{x}} = R^*$$
$$ \quad \quad \...
0
votes
1answer
365 views
Defining a soft constraint in cvxpy
I am using cvxpy to do a simple portfolio optimization.
I implemented the following dummy code
...
2
votes
1answer
99 views
What is the fastest way to solve Ax=b (subject to constraints and an absolute term)
I am trying to solve/optimize $Ax=b$ in the least squares sense subject to
box constraints;
a few (less than 5) equality/inequality constraints; and
an absolute function penalty (or some other ...
1
vote
2answers
69 views
Making difference of log constraints convex
I have the discrete likelihood estimation problem $\max \sum m_i\log p_i $ where $m$ is a given vector of length $n$. The constraints are $0 \preceq p \preceq 1$, $\sum_{i=1}^n p_i = 1, $ and one ...
4
votes
1answer
244 views
Linear constraints for L-BFGS-B
I know L-BFGS-B only supports simple box constrains of the form: $l_i \leq x_i \leq u_i$, where $l_i$ and $u_i$ are constants. For my specific optimization problem, I need to specify some simple ...
0
votes
1answer
55 views
reduced system: primal-dual interior point method for nonconvex constrained problem
When solving a reduced KKT system of a nonlinear (and nonconvex) constrained program after eliminating slack and dual variables, how do we actually take the next step in a primal-dual method?
For ...
4
votes
1answer
99 views
Obtaining a feasible solution for underdetermined system of linear equations satisfying inequality constraints
I would like to obtain a feasible solution for an under-determined system of linear equations,
$$Ax=b$$
where, $A \in \mathbb{R}^{7\times9}, \, x \in \mathbb{R}^{9\times1}\text{and } b\in\mathbb{R}^...
1
vote
0answers
244 views
Optimization of a blackbox function with an equality constraint?
I believe this would be an interesting problem.
I have a blackbox function which can take 2-60 input variables $(X_1,X_2,...X_n)$ which are to be optimized. I'm calling this objective function as a ...
4
votes
1answer
269 views
Ways to speed up solving an LP with Google's ortools
I'm having an issue solving an LP of the form:
$$\min z = c^Tx$$
$$\text{s.t.}$$
$$Ax \geq b$$
$$x\geq p$$
$1 \leq a_{ij} \ll b_i$, $p \leq 0$, and $c \geq 0$
The specific problems I'm running into ...
1
vote
3answers
160 views
Optimization of a blackbox function
Let's say that we have an objective function $f(\mathbf x,\mathbf y)$ which has the parameters $\mathbf x=[x_1\ldots x_n]$ and $\mathbf y=[y_1\ldots y_n]$. Here, $\mathbf y$ is a blackbox variable ...
1
vote
0answers
65 views
Sequential Quadratic Programming for Quadratically Constrained Quadratic Programs
A standard Quadratically Constrained Quadratic Program (QCQP) is of the form:
$$ \underset{x}{minimize} \frac{1}{2}x^TP_{0}x + q_{0}^{T}x
$$
$$
subject \; to \quad \frac{1}{2}x^TP_{i}x + q_{i}^{...
2
votes
2answers
265 views
Knapsack problem with fixed number of elements?
I am looking at an optimization problem that looks like this:
$$
\text{minimize}\;\; \mathbf{x}^TQ\mathbf x \;\;, \; \mathbf x \in \mathbb R^n, x_i \in \lbrace 0, 1 \rbrace\\
\text{subject to}\;\; ||...
4
votes
0answers
43 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{...
1
vote
0answers
92 views
functional second derivative
I'm trying to build a numerical solution for a parameter estimation problem of reaction-diffusion equation, using the adjoint method.
To summarize it, I'm trying to minimize the function
$$ G=\frac{...
