# Questions tagged [optimization]

The process of determining the best solution among all possible solutions given a set of constraints.

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### Lagrange multipliers $\lambda$ in Nonholonomic systems and Costates $\lambda$ in Optimal control theory

In Structure and Interpretation of Classical Mechanics, Section 1.10.3 by Sussman & Wisdom, talking about systems with non-holonomic constraints, it says the following about the Lagrange equations ...
1 vote
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### How to mathematically express and then optimize my photon emitter? [closed]

A laser shoots photons directly at a flat disk with a fixed linear rate f(t) on the interval [t0=a,tn=b] with a horizontal path. E.x. f(t)=100 photons per second The laser is controlled by robotic ...
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### Shape of fastest spinning rod

A one-meter steel rod of variable thickness is attached at one end to a spinning hub. The cross-sectional area of the rod is a function $f(x)$ of the distance $x$ in meters from the hub, x ranging ...
1 vote
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### Does nature perform optimal linear quadratic control?

Given any linear quadratic control problem $$\min \sum_t c_x(x_t, t) + c_u(u_t, t)$$ where $u_t$ is a "control variable" (think of it as an adjustable velocity) and $x_t$ is a "state ...
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### Minimum time passed for stationary observer at destination in relativistic physics

While my friend and I were discussing about relativistic physics, we came up with the following question. Suppose a person $P$ wanted to travel from point $A$ to point $B$ in one-dimensional space. ...
1 vote
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### How to calculate degrees of freedom?

Background I am trying to run optimizations on a multilink (car-) suspension. That is each link is defined by two points, one on the vehicles body, one on the wheel mount. There are 5 links in total, ...
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### How are conjugate variables in mechanics and stat mech related to duality in convex optimization?

I recently studied duality in optimization where a primal optimization problem can be casted as a dual problem which provides meaningful lower bounds on the primal. There is also a notion of conjugate ...
53 views

### Why do we maximize according to the values of Lagrange multipliers?

In some Lagrangian problems, when we use the lagrange multipliers to minimize a function $f(x)$ they write: \begin{equation} \max_{\lambda,\mu} \min_{x}\mathcal{L} = \max_{\lambda,\mu}\min_{x} \Big( ...
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### Given a path and maximum acceleration, what is the minimum time to reach the end?

As stated in the title, I want to find an expression or a way to calculate the minimum time to go from one point of a path to another when the path is given and acceleration is restricted. Thus far, I ...
1 vote
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### 2D boundary value problem for planetary lander [closed]

I'm solving a 2D lander guidance problem where a planetary lander must be guided to a safe landing spot on an arbitrary terrain. The specific method I'm coding use methods similar to RRT/RRT* motion ...
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### Could two concatenated cycloids be an optimal solution to the Brachistochrone problem?

The following is a specific instance of the brachistochrone problem, which I first encountered in grad school, and I have occasionally used as hw problem in teaching CM. A particle is started from ...
1 vote
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### Optimizing engines to produce a certain torque and net force

Say we have $n$ engines sitting on a rigid body. Each engine has position $R_i$ and points in a certain direction and generates a force in that direction, $F_i$. The magnitude of that force ($k_i$) ...
1 vote
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### What methods can I use to find the minimum of a tranverse field Ising model?

I am trying to solve for the minimum of the hamiltonian of the form: $$H = \sum_{i,j} J_{ij}q(i)q(j) + g_i\sum_i x(i)$$ where q(i) is the operator (I + z(i))/2 and z(i) and x(i) are pauli operators ...
1 vote
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### Black Hole Coin curve

I've seen a toy at my local pharmacy called The Black Hole that lets you insert a quarter into a vertical slot. The quarter starts rolling around a curved funnel's circumference and slowly winds its ...
1 vote
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### Minimizing a Line Integral of a Vector Field Using Euler-Lagrange---Problem with Boundary Conditions

Let $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be the following vector field: $$F(x,y)=\left< x\cdot p(x,y(x)), y(x)\cdot p(x,y(x))\right>$$ where p(x,y(x)) = \frac{5x}{6} + \frac{7 x^2}{4} + \...
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### How to minimise sum potential of the system?

Given $m$ number of immobile electric charges at some points of the plane. We want to place $n$ number of further charged particles (not including the fixed charges) along a circle centered at the ...
359 views

### Can quantum computing solve the curse of dimensionality?

The curse of dimensionality is ubiquitous in machine learning (ML) modeling, stochastic control and reinforcement learning, arising in a probabilistic sense, with strong connections to quantum ...
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### How to maximize overlap integral

my question is on finding the function $p(x)$ that maximizes the overlap integral $\int_{-\infty}^{+\infty} f(x)p(x) dx$, where the site condition $\int_{\infty}^{+\infty} p(x)dx=1$ holds (of course ...
128 views

### Why does current arrange itself in such a way as to minimize power loss in resistors connected in parallel?

In problem 2 in Problem Set 6, it is said: "Electricity prefers to flow in the way that minimizes energy loss to resistance." Using Lagrange multipliers I was able to show that assuming the above ...
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### Optimising equation with 3 or more changable variables

In order not to bother you with technical details I've laid my therms in plain math. The sets are experimentally obtained mechanical characteristics, and unknowns are some empirical parameters from ...
4k views

### How to position table legs to distribute weight optimally? [closed]

Background I'm building a table using 4 table legs and a table top. The table top, made out of wood, tends to be quite elastic. Yet, I want the table to be able to hold heavy objects without worrying ...
74 views

### Lagrangian Mechanics to solve arbitrary maximization problem

I've been thinking for some time about how to better find the optimal weights for neural networks and it struck me that when solving Lagrangian mechanics problem you are optimizing the action function....
1 vote
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### Optimal trajectory of rocket with variable specific impulse and constant power

I am trying to derive trajectory of rocket with variable specific impulse and given power of engines $P$ that minimizes the total time. The specific impulse is given as time-variable effective exhaust ...
1 vote
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### Attaining extrema when a stationarity condition has no solution

I was wondering if someone could shed some light on the following for me: If a stationarity (maximizing or minimizing) condition has no solution inside a particular domain, then how do we reason that ...
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### Ising to QUBO mapping for quantum annealing

I'm trying to solve an optimisation problem by simulating quantum annealing using the path-integral Monte Carlo Metropolis approach. So far I have formulated the problem as an Ising model with the ...
1 vote
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### Why Euler-Lagrange Theorem gives the optimal control law? [closed]

Can anyone explain me in simple language why applying Euler-Lagrange Theorem on Hamiltonian equations gives the optimal control law? I am specifically talking about trajectory optimization problems ...
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### Optimally cooling a cup of broth

Say I have a cup of broth with volume V1 and it's too hot to drink. I want to cool it down as quickly as possible. the current temp of the cup is T1 I want to drink the cup when it cools to T2 I have ...
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### Control systems from a physicist's perspective

I am highly interested in the study of control systems theory. However it seems that almost all books are written by electronics or mechanical engineers. Due to this they generally omit many things. ...
1 vote
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### Designing capacitor or resistor systems where capacitor or resistor can bear limited voltage

A capacitance of $2\ \mu F$ is required in an electrical circuit across a potential difference of $1.0\ kV$. A large number of $1\ \mu F$ capacitors are available which can withstand a potential ...
1 vote
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### Numerical optimisation over quantum states/measurements

Suppose we need to find a quantum state and set of measurement basis that optimise a given function of these parameters. This is quite common for e.g. in finding optimal violation of some bell ...
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### Minimizing entropy generation in storing heat

I am working on a cogenerative PV (Photovoltaic) array. The idea is to generate electricity from the solar cells while cooling them to both increase solar conversion efficiency and extract waste heat ...
1 vote
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### Improving classical processors with simulation or optimization on a quantum computer?

Two classes of applications for quantum computers are (1) solving constrained optimization problems [example] and (2) ab initio simulation of quantum systems [review]. Have there been any concrete ...