Questions tagged [optimization]

Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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Given a point in the 3D brain, and given blood vasculature in the brain, find a path to drill through the brain farthest from the vasculature. [closed]

The specific context for this problem is placement of electrodes in the brain. You know the specific point where you want to place the electrode in the brain, and assume you know where the blood ...
Thao Nguyen's user avatar
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Suggested preconditioners for Broydens method?

thanks for reading my post. I am currently using Broydens method to solve a system of nonlinear equations. I identify the initial Jacobian numerically, and then use the Broyden update to identify ...
Josh's user avatar
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2 votes
4 answers
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If in $\triangle ABC$, $r=1,a=3$,then find least possible area of $\triangle ABC$

A circle of radius $1$ unit is inscribed in $\triangle ABC$. If $BC=3$ then find the least possible area of $\triangle ABC$ and also find the perimeter of the triangle when it has the least possible ...
Maverick's user avatar
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If a polyhedron is nonempty and has at least one extreme point, but has no interior), will it have at least one degenerate extreme point?

Background that's helpful: Convex analysis, polyhedra, linear programming, degeneracy, interiors and relative interiors of convex sets, basic feasible solutions of linear programs
Math Music's user avatar
1 vote
1 answer
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Finding the dual of a convex problem with inequality constrains

I practice some exercises from Amir Beck's book "Introduction to Nonlinear Optimization" and I wanna find the dual problem of the primal problem which is given by: $$\text{min } \sum_{i=1}^n ...
Chen's user avatar
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Lower bounds with two paremeters

NB I use the Landeau notations in the following. Also, let us fix some minimization problem and call it $M$. Assume statement $P$ holds true: For any constant $\alpha \geq 1$, an $\alpha$-...
chris765's user avatar
1 vote
1 answer
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$f f'' \geq 2(f')^ 2 \implies f$ decreasing

I'm currently doing a project and trying to find the conditions a function needs to have for the problem to have nice properties. The conditions so far are: $f(x) > 0$, $\; \forall x \in \text{dom ...
William's user avatar
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Optimize Brass Ring Game for Carousel

Background: We are a carousel with a brass ring game. Our game uses a cartridge which dispenses 35 rings per ride, and we have it loaded with 33 plastic rings and 2 brass rings loaded 15-16 rings ...
Carousel Facilities's user avatar
4 votes
2 answers
110 views

Does convergence of gradient to zero imply convergence to optimal value?

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a $C^1(\mathbb{R}^n)$ convex function that has a minimal value $f^\star = \inf_{\mathbb{R}^n} f$. Assume also that $\nabla f$ is $L$-Lipschitz continuous. Let $(...
Kas's user avatar
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Linear programming with different contraint matrix [closed]

I am new to linear programming. When I study this subject at my school, I met an problems, it is as follows:" Prove that if $min${$(c,x) | Ax=b, x \ge 0$} has optimal solution, then $min${$(c,x) |...
Dung Huu's user avatar
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A company is dedicated to the manufacture of three kinds of lenses: A, B and C.

A company is dedicated to the manufacture of three kinds of lenses: A, B and C. The production procedure involves three operations: lens formation, where molten glass is transformed into raw lenses; ...
isa.be's user avatar
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2 votes
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Minimize a quadratic form constrained to the vector being non-negative and sum up to one

I want to solve the following optimization problem with respect to $\mathsf{x} = (x_1, \ldots, x_n)^\top$ $$ \min_{\mathbf{x}} \,\, \mathbf{x}^\top A \mathbf{x} \qquad\qquad\text{such that }\,\, x_i \...
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1 vote
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Design a subspace to contain the most possible of some vector

Say $P_X$ is the projection onto $X$'s image. Fix $v\in \mathbb{R}^n_+$ and $y\in \mathbb{C}^{k+n}$. Now what is the largest possible value for $y'P_Xy$ among those $X\in \mathbb{C}^{(k+n)\times k}$ ...
Christian Chapman's user avatar
1 vote
1 answer
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show that strong duality holds

I try to show that for the given problem, strong dualty holds: Primal Problem: $$ Min \text{ x,y}: \sqrt{\lVert x \lVert^2+4}+a^T y+\lVert x \lVert $$ $$ \text{s.t }\\ Bx+Cy \leq d \text{ , }\lVert ...
Chen's user avatar
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2 votes
1 answer
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How to Maximize $\sum_{k=1}^{n} \frac{M_{k}}{m_{i} - \sum_{p=1}^{k} M_{p}}$

This is from a physics problem : A rocket with initial velocity $v_{i}$ and initial mass $m_{i}$ shoots $n$ pellets. The final mass and velocity of the rocket is $m_{f}$ and $v_{f}$. Determine the ...
Vue's user avatar
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1 vote
2 answers
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determine whether a function is coercive or not [closed]

For the next exercise, I need to check whether the next function is coercive, the function is of the form: $$f(x,y)=x^2 -2xy^2+\frac{y^{4}}{4}. $$ So far I tried to square it and bound it from below, ...
Chen's user avatar
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1 vote
1 answer
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Pair-wise assignment but neighbours can't be paired together

I have a set of N elements and I would like to assign them pairwise using a function f(n) where ...
x41lakazam's user avatar
1 vote
1 answer
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what is wrong with my equivalent transformation on optimization below

$\mathbf{R_1}$ and $\mathbf{R_2}$ are positive definite symmetric matrices, $\mathbf{q}\in\mathbb{C}^{N\times1}$, and I want to maximize the objective function below $$ J(\mathbf{q})=\frac{\mathbf{q}^{...
lei zhou's user avatar
1 vote
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Is it possible to connect any point $a$ to a global minimum $b$ for $f$ with a path $\gamma$ such that $f\circ\gamma$ is non-increasing?

I am doing some research on PDE theory, and in order to prove a problem I'm working on, I had an idea, and this idear relates to the following problem Let $E$ be a Banach space and $f:E\to\mathbb{R}$ ...
Jacaré's user avatar
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1 answer
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Proof of Coerciveness of a Function [closed]

please may have a look . Help, Hints, $\textbf{Statement:}$ Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^1$ function such that for every $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$, the ...
E.K's user avatar
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Solve Minimax Rules in Finite Case

Let $\Theta = \{\theta_1, \cdots, \theta_n\}$ be the space of parameters and $D = \{d_1, \cdots, d_m\}$ be the space of decisions (that is, they are arbitrary finite sets with at least two elements). ...
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Optimization Problem for cylindrical cans [closed]

A manufacturer wants to make cylindrical cans. Each can must be able to hold 1000 cubic cm of liquid. Find the dimensions that minimize the cost of metal to be used to make each can. What is this cost ...
Kim082's user avatar
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1 answer
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Minimal angle between Euclidean subspaces

Consider the following minimization problem. Let $U, V$ be $k$ and $l$ dimensional subspaces of $\mathbb{R}^n$, respectively, so that $k+l \leq n$, and $U \cap V = \{\textbf{0}\}$. Suppose $\mathbb{...
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Does Cutting Plane method always converge in case of Integer Linear Programming?

I learned that the Cutting plane algorithm using Gromory's cut helps in finally reaching an optimum solution in integer linear programming. But I also observed that in the simplex tableau, if the ...
Vinny Dek's user avatar
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1 answer
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Solving $Ax=b$: Projection onto subspace with a canonical basis of largest error

The goal is to solve the linear system $Ax = b$, where $A$ is symmetric and positive definite (SPD). Consider the one-dimensional projection method given by equation (1): $$x_{k+1} = \operatorname{...
Meow's user avatar
  • 165
2 votes
1 answer
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Minimum of the dot product over the vertices of a regular n-simplex

For $n \geq 2$, let $\Delta^n$ be a regular $n$-dimensional simplex in $\mathbb{R}^n$ centered at the origin $0$ and inscribed in the unit sphere $\mathbb{S}^{n-1}$. Let $v_0,\dots,v_n \in \mathbb{S}^{...
Paul Tristant's user avatar
2 votes
2 answers
58 views

Maximise $f(x,y)=x^2+y^2$ on contraint that looks like infinity sign

I would like to find the maximum of the function $f(x,y)=x^2+y^2$ on the constraint $x^2-y^2=(x^2+y^2)^2$. The level curve $h(x,y)=0$ of the function $h(x,y)=x^2-y^2-(x^2+y^2)^2$ looks like an ...
Apollo13's user avatar
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2 answers
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Identify optimal product size configuration based on historical data and some constraints [closed]

We have historical data for the demand of a product. Product can be demanded in any quantity between 0-1000g and the historical data show the distribution of previous request sizes. We can only pack ...
user896201's user avatar
4 votes
0 answers
94 views

Procrustes with inequality constraint

I am interested in a variant of the orthogonal Procrustes problem $$ \begin{array}{ll} \underset{X} {\text{minimize} } & \| X A - B \|_{\text F} \\ \text{subject to} & X^TX = I \\ & (X^T ...
xdaimon's user avatar
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0 answers
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Simple Optimization Problem from Economics

An individual has 100 dollars that they have to invest in Asset 1 or 2. The returns from the two assets are $r1\stackrel{d}{=} c + v_1, r2=v2$ for some constant $c$ and independent random variables $...
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-2 votes
0 answers
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Minimizing Distance Between Vectors Through Linear Transformation

I'm working with two vectors, $Y$ and $X$, and I have to find a approximation for $Y$ using a linear transformation of $X$. I'm looking to find scalars $a$ and $b$ that minimize the distance between $...
Peyman's user avatar
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1 vote
1 answer
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Why this ILP and LP are equivalent?

Let's consider a competition with $n$ questions. Each question has a price $p_i$ and a score $v_i$. To advance to the next round of the competition, we need to accumulate a minimum score of $D$. We ...
occasional's user avatar
0 votes
1 answer
17 views

Proximal operator of the Moreau envelope

I'd like to know if there is known general formula for computing the proximal operator of the Moreau envelope of a given suitable function. Given the following definition of Moreau envelope $$ M_{\rho,...
Gabrio's user avatar
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0 answers
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Feasible set formed by exclusion of two convex sets

I'm working on an optimal control problem which is almost entirely composed by elements of a quadratic programming problem. The decision variable is $u \in \mathbb{U} \subseteq \mathbb{R}^2$, where $\...
Lucca's user avatar
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1 vote
1 answer
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Optimize combination of resources to achieve multiple objectives

I have a bunch of resources that each yield a combination of outputs, and I’m trying to find the optimal amount of each resource to achieve a minimum amount of each output. Simplified Details: ...
wasystr's user avatar
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1 vote
1 answer
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"Sandwich" Quadratic majorizer of even convex function

Assume $f : R \to R$ is strictly convex, differentiable, even and $f(0)=0$. Further assume that there exist quadratic functions above f. I want to prove that for any quadratic function $ax^2 + bx + c \...
test-account's user avatar
0 votes
1 answer
22 views

Choosing k elements with multiple weights maximizing the minimum weight

Consider the following optimisation problem. Given a set $S$ with $q$ weight functions $w_1, \ldots, w_q: S\rightarrow \mathbb{R}_+$ and a constant $1\leq k\leq |S|-1$. Find an $X\subset S, |X|=k$ ...
Bence's user avatar
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1 answer
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Best root finding algorithm to use for this problem..

The following formula is used to find the strike, $K$, that achieves the maximum delta on a premium adjusted FX call option priced under black-76 model. $$ g(K) = \sigma \sqrt{T} N(d_-(K))- n(d_-(K)) =...
Attack68's user avatar
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0 votes
1 answer
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parametric optimization in differential equation through gradient descent

I need to solve a differential equation containing a parameter $n$, and based on a constraint I assign, I would like to find what is the value of $n$. I would like to use a gradient descent method. ...
Marco Gandolfi's user avatar
5 votes
0 answers
71 views

Find a maximal area of a convex figure whose all $\mathbb{Z}^{2}$ shifts can make a full circle turn (current record is 0.8064846)

Problem. Consider the plane $\mathbb{R}^{2}$. Given a convex solid figure $\mathcal{P}$ such that $(0,0)\in\mathcal{P}$. For every pair $(n,m)$ of integers, let $\mathcal{P}_{n,m}$ be a shift of $\...
Giedrius Alkauskas's user avatar
0 votes
1 answer
22 views

Relationship between proximal mapping and subgradient

Background: I came across this excerpt on Wikipedia Can anybody please help in computing exactly how $$0\in \delta(\lambda f(z))+(z-x)$$ $$\Leftrightarrow 0\in \delta(\lambda f(z)+1/2 \lVert z-x\...
mtcicero's user avatar
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0 answers
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Combinatorial Optimization Assignment Problem as Graph Coloring Problem [closed]

Im trying to present a Combinatorial Optimization Problem that is kind of Assignment-Like in a different way so that it is perhaps easier to solve with conventional Algorithms. I'd like you to look ...
ImNotSure's user avatar
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0 answers
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Black Box Curve Fitting Problem with Extra Layer Between Input and Output Vectors

MAJOR EDIT: I attempted to better describe my problem here. My original post (more or less) is below, if for whatever reason that is needed. $\underline x$ is a linearly spaced vector of arbitrary ...
kriegersan's user avatar
-1 votes
0 answers
31 views

Maximization problem for N circles arrangement, each having its diameter inside another [closed]

We are given a circle of diameter 1 and centered at origin, we need to find the best arrangement of $N-1$ more circles, each having its diameter inside previous one (circle number $j$ has its diameter ...
Vladimir_U's user avatar
0 votes
1 answer
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Does a convex quadratic program have a unique dual solution?

As shown in Does a convex quadratic program have a unique solution?, a convex quadratic program has a unique primal solution $x$ if $Q$ is PD. $$\min \; x^T Q x \\ s.t. \ Ax= b : \lambda \\ \ \ \ \ \ \...
Sean W's user avatar
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4 votes
3 answers
175 views

Geometric approach to minimize inner product

Problem : Let $S$ be a surface $$S : z = \frac{x^2}{4} + \frac{y^2}{2} + 1$$ For two point $Q(x,y,z)$ on $S$ and $P(2,0,1)$, evaluate $$\min \frac{\mathrm{OP}\circ\mathrm{OQ}}{|\mathrm{OQ}|}$$ My ...
bFur4list's user avatar
  • 2,636
2 votes
0 answers
34 views

Changing convexity by changing metrics

I was wondering if for a given functional $F$ on some space $X$, is it possible to construct explicit metrics $(X,d_1)$ and $(X,d_2)$ such that $F$ is convex w.r.t $d_1$ while $F$ is non convex w.r.t $...
Silentmovie's user avatar
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0 answers
27 views

How to find function $G(v_i)$, such that $\mathbb{P}\big[{\psi}(G(v_i))\big]= \frac{1}{G(v_i)}$. Is this a fixed-point problem?

The problem comes from an economic and market scenario. We have a function $$\psi(v_i)=v_i-\frac{G(v_i)-F(v_i)}{f(v_i)},$$ where the random variable $v_i$ is any real number (e.g., person $i$'s money)...
iDhone's user avatar
  • 1
2 votes
1 answer
48 views

Tractable formulation of a mixed integer program

Given constant matrices $A_1\in\mathbb{R}^{1\times l}$ and $A_2\in\mathbb{R}^{1\times l}$, and constants $b_i$, $i=1,\dots,n$. Consider the following mixed integer program (MIP) with decision ...
Jeremy's user avatar
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1 vote
0 answers
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Determining Optimal Policy Probabilities in Off-Policy PPO Using KKT Conditions

I'm working through a paper on Proximal Policy Optimization (PPO) and am trying to understand the derivation of the optimal policy probabilities for the off-policy case as expressed in Equation 16. &...
Amantuer Rewuhan's user avatar

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