# Questions tagged [constraints]

In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy.

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### Name of this parametrization of the region $\{ 0 < x \leq y \leq z \}$

I'm working on a problem where I have to assign a probability density $p(x,y,z)$ on $\mathbb{R}^3$ which is non-zero only on the region $$R(x,y,z) = \{ 0 < x \leq y \leq z \}.$$ To this end I use ...
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### 10 similar quadrilaterals with 8 points.

Let 8 points define 10 or more similar quadrilaterals with no self-mapping. No self-mapping means that a square counts just once instead of 8 times. What is the maximal number of differently sized ...
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### Distribution of random numbers with fixed sum

I have access to a black box function $f$ that returns 4 random integers $n_1$, $n_2$, $n_3$, $n_4$ with $4 \le n_i \le 13$ and $\sum_i n_i = 25$. Experimentally, I can see that $n_1$, $n_2$, $n_3$ ...
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### Constraint Optimization and Lagrange Multipliers (Methods of Optimization)

Newbie question here. So I am starting to learn about constrained optimization in my multivariable calculus course and I was taught how to use the Lagrangian and Lagrange multipliers to solve an ...
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### Drawing Level Curves and Gradients on the same graph

$g_1(x,y) = x^2 - y$ , $g_2(x,y) = y$ , and $g_3(x,y) = x$, Firstly I drawed the level curves corresponding to $g_1(x,y) = 0$, $g_2(x,y) = 0$, and $g_3(x,y) = 1$. Then, I shaded the region all points ...
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### Show set of solution to constraint is compact

This is the question: Show that the function $f(x,y) = x^4 + y^4$ takes maximum and minimum values along the curve $x^4+y^4-3xy = 2$. A solution from my teacher: Since $f$ is continuous, we only ...
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### Understanding the adjoint (state) method: existence and uniqueness of the adjoint equation

Dear math enthusiasts, I recently came across the adjoint (state) method in the context of sensitivity analysis of model perturbations to systems described by PDEs. I am a novice in the area so I was ...
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### Loss Function vs. Penalty vs. Lagrange Multiplier

Suppose I want to maximize $u=c_1+c_2$ subject to a budget constraint $p_1c_1+p_2c_2\leq m$ and a requirement $c_2<\overline{c_2}$ that must be satisfied. The source of the $c_2<\overline{c_2}$ ...
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### Constrained optimization problem with amplitude constraints

My problem is based on beamforming. I want to minimize the consequence of $\|Hw\|_2^2$, where $Hw\in \mathbb{C}^{N \times 1}$, is the power of signal coupled to receive antennas from the transmit ...
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### Monotonic increasing function with constraints

I am looking for examples of monotonically increasing functions f(x) which can satisfy the following constraints: f(x=0) should be 0 and f(x>=25) should be 1. I want one or more parameters in the ...
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### Constructible problems for which the solution is non-constructible?

For the sake of this question, I am using the word "constructible" in the sense of constructive mathematics: e.g. a real number is constructible if you can construct a Cauchy sequence for it ...
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### Independent conditions of unitary matrix, $U U^{\dagger}=1$ and $U^{\dagger}U = 1$

I am trying to show that, for a simple 2x2 complex matrix \begin{equation} U = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \end{equation} there will be only 4 real constraints for $U$ to be ...
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### How to relax logical constraints

Consider two $m\times 1$ vectors, $x\equiv (x_1,x_2,...,x_m),\tilde{x}\equiv (\tilde{x}_1,\tilde{x}_2,...,\tilde{x}_m)$. Let $x\leq \tilde{x}$ if and only if $x_i\leq \tilde{x}_i$ for each $i=1,...,m$....
So I have the following optimization problem \begin{aligned} \max_{x:=(x_1,\dots,x_n)^T\in\mathbb{R}^n} \sum_{i=1}^n x_i \\ \text{s.t. }\quad \text{ det} G(x) = 0 \end{aligned} were $G(x) = I+\... 0answers 43 views ### Solve the inequality$ a^{ab}c+b^{bc}a+c^{ca}b\leq 1-2abc$Let$a,b,c>0$such that$a+b+c=1$then we have : $$a^{ab}c+b^{bc}a+c^{ca}b\leq 1-2abc\quad (1)$$ I have a proof : I was thinking for an alternative proof considering by example Young's ... 0answers 9 views ### Build a heart like curve following some constraints Hi it's a funny problem . Working on some curve I have found a Heart's curve let me propose it . There is four curves to define the heart's curve .First : Let$0< x< 1then define : f(x)=x^x+(... 1answer 42 views ### Maximize sum of logarithms subject to constraints I have the optimization problem \begin{align} \: \max \: \sum_{i = 1}^{M} \log_2\left(1 + \frac{S_{i}}{N_i}\right) \\ \text {Subject to} \: \sum_{i = 1}^{M} S_{i} \leq P_T \end{align}. N_i are ... 0answers 23 views ### How to linearize a non-linear constraint in an optimization problem? I am very new to optimization, and I wouldn't know where to begin with this problem that I have so I would truly appreciate any and all help on this - I am trying to find a vector \boldsymbol{x} = \... 0answers 13 views ### General algorithm for addition of two constrained numbers I am looking to write a general algorithm for addition of constrained numbers and am wondering if someone could give me some guidance on where to start or if there is already a paper on strategies for ... 0answers 10 views ### Question about Constrained Differential Solution https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-c-lagrange-multipliers-and-constrained-differentials/session-42-constrained-differentials/... 1answer 38 views ### How to find a closest point on a constraint to a given point? (This is a multi-variable calculus problem, not a linear algebra one) Find the closest point to the point (2,7,8) on the constraint: 4x + 7y = x+3y+5z$$And find the projection of the vector (2,... 0answers 17 views ### How to obtain a rank-5 solution to an optimization problem Say I want to determine a matrix solution A^* to the minimization problem$$\min_{A \in \mathbb{R}^{n\times n}} f(A)$$with the constraint that this solution A^* must have rank k, where k \ll n... 4answers 83 views ### Find the maximum value of x^2y given constraints Find the maximum value of$${ x }^{ 2 }y$$subject to the constraint$$x+y+\sqrt { 2{ x }^{ 2 }+2xy+3{ y }^{ 2 } } =k$$where k is a constant. I tried it by substituting value of x and then ... 1answer 28 views ### Expressing inequality constraints as equality Is it possible to express inequality constraints as equalities? I have a system of linear equations that I am trying to solve where the system is subject to a set of inequality constraints. The ... 0answers 32 views ### How to constrain a rectangle within an arbitrary 2d polgyon? I am working on a graphics project wherein I need to keep a rectangle (assume it has a fixed but arbitrary size, rectWidth * rectHeight) constrained inside an arbitrary polygon. The polygon is ... 1answer 26 views ### How to solve an objective function with inequality constraint? I have an objective function as follows, \displaystyle\min_{\mathbf{A},\mathbf{B}}\lVert \mathbf{X}-\mathbf{ABZ} \rVert_F^2, where \mathbf{X},\mathbf{Z}\in \mathbb{R}^{p\times n}, \mathbf{A}\in ... 1answer 35 views ### constrained optimal control excluding the optimal state I don't know if this is a very basic question: Let's say there is the typical optimal control problem with the cost function$$J = \int_{0}^{T} \mathcal{L}(x(t),u(t),t)\mathrm{d}t + \Psi(x(0),t=0)$$... 0answers 42 views ### Closed form for the maximum of the two-variable function (a x + (1 - a) y) (b (1 - x) + (1 - b) (1 - y)) I'm trying to characterize the maximum of this function within the unit interval (x,y\in [0,1]):$$f(x,y)=(a x + (1 - a) y) (b (1 - x) + (1 - b) (1 - y))$$for 0 < a < 1 and 0 < b < 1... 0answers 43 views ### Efficient Method to Find A Root and/or Minima of A Multivariate Polynomial With Constraints on the Variables I have a polynomial P(x,y,z,...) which has integer coefficients and real roots. I want an efficient method that is guaranteed to find a root (I only need one), and if possible, one that finds a ... 1answer 98 views ### Identify boolean function that satisfies some constrains The problem I want to find a boolean function f(x,y):\{0,1\}^n \rightarrow \{0,1\}, where x=\{x_i\}_{i=1}^{m} and y=\{y_i\}_{i=1}^{k} are m and k boolean variables, such that: m,k \ge 1 (... 1answer 29 views ### If evey linear program can be transformed to an unconstrained problem, then the optimum is unbounded because the objective is linear? Since optimization problems with linear equality constraints can be converted into an unconstrained problem this should apply for linear programs in standard form, right? But doesn't this mean that ... 1answer 53 views ### Maximization of quadratic form subject to a set constraint Given a quadratic form x^tAx where the matrix A is symmetric and x^tx = 1, we can deduce that the maximum of the quadratic form is the first eigenvalue of the matrix A (also, the first ... 0answers 37 views ### What constraint am I missing for this SAT optimization problem? I'm trying to solve a variation of the bin-packing problem. I have a certain floor space and I wish to place as many boxes as I can without stacking, i.e. if the floor is a 4 x 4 grid and I have one ... 1answer 19 views ### Alternative definition of an active constraint In constrained optimization, an active constraint is generally taken to mean one whose inequality sign can be changed into an equality sign without affecting the optimum. However, I would like to have ... 1answer 84 views ### Lagrange Multipliers and quasi-Newton methods Consider an optimisation problem of the form$$ \begin{aligned} &\min f(x)\\ &\text{s.t. } g(x) = 0 \end{aligned}$with$f,g: \mathbb{R}^n \to \mathbb{R}$convex and twice continuously ... 2answers 24 views ### Optimization constraints for types of oil I'm looking at a lecture online which shows the following problem They have two types of oil and they want to produce standard and premium gasoline. However i'm confused how they've arrived at the ... 1answer 36 views ### Safe packing Constraint satisfaction problem - is it optimal? Problem: You need to pack several items into your shopping bag without squashing anything. The items are to be placed one on top of the other. Each item has a weight and a strength, defined as the ... 0answers 12 views ### Efficient intersection of linear subspaces? How to solve a big nonlinear least squares with linear constraints? Problem: I want to solve a big linearly constrained nonlinear least squares problem. The number of unknowns is between millions and billions. In terms of cost functions, the Hessian would be pretty ... 0answers 20 views ### How to verify whether a optimization problem with a L2 constraint (quadratic constraint) and a linear constraint has its closed-form or not? I have already known that the optimization with a quadratic constraint can be solved using Rayleigh quotient, and with a linear constraint can be solved by Lagrange method. But what if the two ... 0answers 19 views ### Existence of solutions to constrained inequality Consider a system of$N$points in 3D Euclidean space (although it would be cool if it can be analyzed in general dimensionality) with weights$0 < w_i \leq w^{\bf{max}}$and pairwise separations$...
Let $d\in \mathbb{R}^n, \ b\in \mathbb{R}^n, \ A \in \mathbb{R}^{m\times n},\ \lambda\in \mathbb{R}$. Let $x=\lambda d+\varepsilon$, where $\varepsilon\in \mathbb{R}^n$. Let \$E_\lambda =\left \{\...