# Questions tagged [constrained-optimization]

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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 ... 0answers 94 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.$Ais small size (e.... 0answers 317 views ### Optimization on the manifold of stochastic matrices So I have an optimization problem of the form $$\text{maximize}\hspace{3mm}f(A):{\bf R}^{K\times K}\rightarrow{\bf R}$$ $$\text{subject to}\hspace{19mm}A^T{\bf 1}=\bf{1}$$ $$\hspace{33mm}A\geq 0$$ ... 0answers 62 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 ... 0answers 76 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(... 0answers 50 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{... 0answers 158 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: $$\frac{\... 0answers 447 views ### Best way to add a positivity constraint to Newton's Method So given an objective function f({\bf x}), I would like to include a positivity constraint when I perform the fixed point iteration:$${\bf x}^{(t+1)}={\bf x}^{(t)} - \text{H}_f^{-1}\nabla f({\bf x}^... 0answers 153 views ### Eigenvalue-style optimization with quadratic constraints SupposeA\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 ... 0answers 27 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 ... 0answers 20 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 ... 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 ... 0answers 70 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... 0answers 92 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 (... 0answers 199 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 ... 0answers 74 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,... 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, ... 0answers 159 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 ... 0answers 72 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/... 0answers 72 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 ... 0answers 40 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) = 0x \in X, y \in Y$$where X and Y are compact convex sets, g(x) and f(y) are ... 0answers 52 views ### How to find two points within defined region in this constrained optimization problem? I am doing a project related to robotics where I am using fmincon function from matlab to minimize the distance between the points ... 0answers 40 views ### Sensitivity Lagrangian solution general case I have asked this question already on maths and mathoverflow. Just a question about a literature reference. I am writing a paper for engineers. Usually for the Lagrange multiplier problem$$ \... 0answers 269 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 ... 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 ... 0answers 94 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{... 0answers 206 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 "... 0answers 41 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 ... 0answers 63 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 $$F^h[\mathbf{x}^h(\... 0answers 335 views ### Solving constrained BVP, singular Jacobian The boundary value problem is$$ \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) =... 0answers 91 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 ... 0answers 117 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 ... 0answers 48 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 ... 0answers 64 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$A$... 0answers 80 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-... 0answers 129 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}... 0answers 63 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)... 0answers 11 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, ... 0answers 34 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 ... 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 ... 0answers 35 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))... 0answers 44 views ### Space covering optimization I have the following problem: In the spaceE=\{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 ... 0answers 84 views ### Sequential Quadratic Programming for Quadratically Constrained Quadratic Programs A standard Quadratically Constrained Quadratic Program (QCQP) is of the form: $$\underset{x}{minimize} \frac{1}{2}x^TP_{0}x + q_{0}^{T}x$$ $$subject \; to \quad \frac{1}{2}x^TP_{i}x + q_{i}^{... 0answers 97 views ### functional second derivative I'm trying to build a numerical solution for a parameter estimation problem of reaction-diffusion equation, using the adjoint method. To summarize it, I'm trying to minimize the function$$ G=\frac{... 0answers 74 views ### Find a vector B that minimizes |W-A*B| I want to find a candidate vector$Bthat $$\min|(W - A_i * B_i)|$$ $$a_i > 0,\ A_i=\{a_0,...,a_i\},\ B_i=\{-1,0,1\}^i$$ For example, given $$W = 0.6,\quad A_4 = [0.1, 0.2, 0.4, 0.7]$$ one ... 0answers 53 views ### Obtaining the lagrangian multipliers in an optimization problem Suppose we have this simple optimization problem \begin{align*} \underset{x\in V}{\text{min}} &~ f(x) \\ \text{s.t.}& ~x \leq \beta \end{align*} Using slack variables \begin{align*} ... 0answers 68 views ### What are the numerical properties to consider between Augmented Lagrangian and the Penalty Method? I'm interested in (locally) minimizing a smooth nonconvex objective function: $$f(\textbf{x}_1, \textbf{y}_1,\cdots, \textbf{x}_n, \textbf{y}_n)=\sum_{i=1}^ng(\textbf{x}_i, \textbf{y}_i)$$ Subject ... 0answers 86 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 ... 0answers 77 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 projectionP(x)\$. ...
I have an objective function that I can write either in quadratic programming (QP) such as $$\sum_{i=1}^N \sum_{j=1}^N C_{ij}^2$$ or as an LP problem $$\sum_{i=1}^N \sum_{j=1}^N |C_{ij}|$$ which can ...