Questions tagged [optimization]
The process of determining the best solution among all possible solutions given a set of constraints.
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About the form of the cost functional for quantum optimal control theory
In quantum optimal control theory, the cost functional is often defined as (e.g, see Eq.(9) in here, as well as many other solid references such as this):
$$J = \langle \psi(T) \rvert O \lvert \psi(T) ...
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Reformulating problem into form of Ising Hamiltonian
The Ising Hamiltonian has the following form:
$$H= -\sum_{j<k}J_{j,j+1}\sigma_j\sigma_{j+1}-\sum_{j} h_j\sigma_j + \varepsilon,$$
Where $\sigma$ are the spins that take values of $\pm$ 1
I have a ...
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Tipping a cylinder [closed]
A horizontal force is applied to the top of a cylinder, creating a torque on it trying to tip it over. Now I have wondered to what height should the cylinder be filled with water to render it most ...
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Rocket propulsion energy efficiency
What ratio of final to initial mass of a rocket to achieves the highest energy efficiency - the highest ratio of final mass kinetic energy to chemical energy expended? And more generally the relation ...
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Mapping an arbitrary spin graph to one with nearest neighbour interactions
I remember having heard once that generic spin-graphs e.g. Ising, or at least 2-local ones (defined as the Hamiltonian contains pairwise interactions at most), can always be mapped to one another one (...
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Difficulty solving conformal-bootstrap-like crossing equations using semidefinite-programming (SDP) via SDPB software
My question involves semidefinite programming (SDP) in the sense of attempting to find some vector $\alpha^{\mu}$ that satisfies the following conditions:
Normalisation: $\alpha^{\mu}n_{\mu} = 1$
...
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How to obtain the goal function of DMRG by Lagrange multiplier method?
The goal of DMRG is to minimize the expectation value of energy, which can be written as
$$ \frac{d}{d |\psi \rangle} \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle} = 0. \...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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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Optimizing a Capacitance function
I am trying to find the optimum values, in order to maximize the following equation:
$$ C (L, (b/a)) =\frac{L 2\pi k\epsilon_0}{\ln(b/a)} $$
where
$$ \frac{dC}{d(b/a)} = -\frac{L2\pi k\epsilon_0 \ln(b/...
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Can the energy of a physical system be described as an unconstrained optimisation problem?
Sorry if this is something that is well known, not really familiar with modern physics beyond high school / introductory undergrad level. I largely work in deep learning and broadly speaking, you can ...
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Brachistochrone to a vertical line [closed]
Just for fun, I am working through some problems in Mathematics of Classical and Quantum Physics by Byron and Fuller. Problem 2.13 reads:
Prove that a particle moving under gravity in a plane from a ...
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Application of KT?
I am not familiar with physics and I am lost about how should I actually apply this to the result from the computer simulation.
I would like to apply 𝑘𝐵𝑇 to the results of the computational ...
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Convex optimisation for holomorphic functions [closed]
Convexity/concavity are useful properties of a function when performing optimisation, as one can rest assured that the found solution is the global minimum/maximum. For a real function of one or two ...
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Optimal control of a damped harmonic oscillator through a lossy, dispersive medium
Say a damped harmonic oscillator is at the far end of a dispersive, lossy medium governed by, say, Debye relaxation. I would like to determine what the optimal arbitrary input waveform is to obtain ...
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Maximum horizontal distance of a freefalling ball with one allowed bounce anywhere along its initial path?
I was washing a spoon in the sink and this question popped into my mind:
If a ball is dropped from height H and is allowed a single deflection of any angle $0 < \theta < 180$ at any height $ 0 \...
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For discrete optimization problems, why does continuous approximation - solving - discretization work?
I was wondering about the following question.
In maths/computer science, many optimization problems ask for integer optimizations. A prominent example would be spin-glass (Ising) systems for simulated ...
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Variational method: Why do parameters differ for two trial functions (optimization)?
Below the potential and trial functions:
$$V(x)=(x^2-1)^2-x^2$$
Use the variational method with the two trial wave functions:
$$\psi_{\pm}(x)=A\left(e^{-\frac{(x-x_0)^2}{2\sigma^2}}\pm e^{-\frac{(x+...
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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 ...
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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 ($k_i$) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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....
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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Maximising velocity at B when rolling down the curve between A and B
I would like to build a curve between two points A and B. A ball would roll down the curve in a gravitational uniform field (i.e., I'm actually going to build the thing here on Earth).
My question ...
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Quantum Approximate Optimization Algorithm
I try to understand the 'Quantum Approximate Optimization Algorithm' (QAOA) by Farhi et al. - arXiv:1411.4028.
I understand that the solution is hidden in the unitaries, but I do not understand how ...
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