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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Finding extreme values of a variable on an intersection of a sphere and a plane

Determine the minimum and maximum value of the variable $z$ defined by the curve given by: \begin{cases} x^2+y^2+z^2=1 \\ x+2y+2z=0 \end{cases} So do I need to find a function $z=f(x,y)$ or just ...
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0answers
16 views

Gradient w.r.t. boundary conditions in PDE

I am trying to solve the following problem. Suppose I have a field $\Phi(r)$, which is the solution to a partial differential equation: $\mathcal{L}\Phi(r) = s(r)$, as long as $r \neq r_0$ Here ...
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0answers
23 views

maximum area under the function, with length constraint

Suppose I have a function $f$, such that $f(0) = f(1) = 0$. Given the length $l$ of the curve between $0$ and $1$, which function maximizes the area under the curve? I know that if $l \leq \pi/2$ the ...
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1answer
16 views

maximum of function in bounded area

How can i calculate maximum of $ \frac{-1}{(x+y+3)^{2}} $ in [-1 1]x[-1 1] with non numeric method. I know that -0.2 is maximum of this function with numeric method and The Hesian matrix is zero . ...
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0answers
8 views

Holder's inequality/Cauchy-Schwartz for Bregman Divergence?

Consider the Bregman divergence. $$ D_F(p, q) = F(p)-F(q)-\langle \nabla F(q), p-q\rangle. $$ And its dual norm: $D_{F*}(p, q) $ where $ F^*(y) = \arg\min_x \left\{ \langle x, y\rangle - F(x) ...
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1answer
13 views

Optimal subset problem

Given a finite set: $X=\{x_1, x_2, ..., x_n\}$, $x_i \in U$, where $U$ is some universe of objects, define: $f:P(X) \rightarrow {\rm I\!R}$, where $P(X)$ is the powerset of X, to be some function. ...
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1answer
17 views

Prove that $\int_{0 \le u \le 1,\Omega}g^2(x)udx$ in term of $u$ is convex

I am having a cost function and I want to know whether convex or not. Could you explain help me my problem? My problem is that given a cost function such as $$F(u)=\int_{0 \le u \le ...
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0answers
48 views

Help with Linear Algebra Optimization Problem. 4 people crossing a bridge

"Four people, A, B C and D need to get cross a river. The only way to cross the river is by an old bridge, which holds at most 2 people at a time. Being in dark, they can not cross the bridge ...
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2answers
29 views

Find the maximum and minimum of the function $f$

Find the maximum and minimum of $f(x, y)=xy-y+x-1$ at the set $x^2+y^2\leq 2$. I have done the following: Since the region $x^2+y^2\leq 2$ is closed, $f$ has a maximum and a minimum, which is ...
2
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1answer
32 views

Proof of convergence for the proximal point algorithm

I'm trying to come up with a super simple proof of convergence on the proximal point algorithm, which uses the iteration scheme $x^{i+1} = \mathbf{prox}_{tf}(x^i)$ where $\mathbf{prox}_{tf}(z) = ...
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1answer
35 views

How to hedge a sports bet

Suppose I've got a $200 ticket on the Golden State Warriors to win the NBA Finals at 5 : 1. The finals start next week, with the Cavs listed at 2 : 1 to beat the Warriors and the Warriors 4 : 9 to ...
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1answer
30 views

Algorithm - Maximum subarrays with sum and OR

I was thinking on the following problem: Given an array A. The value of an interval from i to the index j is defined as follows: Take the maximum value from that interval, and add it to the OR ...
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1answer
18 views

What would be the objective functions for this problem?

I have the following data (this is just a sample of my entire dataset): # Distance PriceIndex Rating 1 400 3 5 2 420 2 4 3 500 1 2 Considering the ...
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0answers
13 views

Absolute value equality of variables in optimization problem [on hold]

I am trying to solve an optimization problem using either MATLAB's built-in $linprog$ function, or with MATLAB $CVX$ front-end. These tools provide an easy way to model constraints such as $A \cdot x ...
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0answers
20 views

Find the extremas of the fuction $f$

I have to find the extremas of $f(x, y)=3x+2y$ subject to $2x^2+3y^2 \leq 3$. Since the region $2x^2+3y^2 \leq 3$ is closed, $f$ has a maximum and a minimum, which is either at the boundary or at ...
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1answer
40 views

Theorem of Lagrange multipliers - Extremas of $f$

I have to find the extremas of $f(x, y, z)=x+y+z$ subject to $x^2-y^2=1$, $2x+z=1$. I have done the following: We will use the theorem of Lagrange multipliers. The constraints are ...
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0answers
20 views

Linear Algebra L2 minimization

Im really confused about how to solve this question or even what its asking. Any help would be much appreciated! Let A be an m x n real matrix ($m \gt n$). Let x* be the minimizer of $||Ax - b||^2 + ...
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1answer
16 views

Shortest Path with Constraint

What is the length of the shortest path that goes from $(0,2)$ to $(12,1)$ that touches the $x$-axis? I tried using calculus to solve this problem (i.e.: distance is: $$ \sqrt{(x-0)^2 + (0-2)^2} + ...
3
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1answer
51 views

An Extension to the Generalized Eigenvalue Problem

Given two square matrices $A_1,A_2 \in \mathbb{R}^{n\times n}$, the generalized eigenvalue problem is finding the scalar $\lambda \in \mathbb{C}$ and vector $x \in \mathbb{C}^{n}$ such that $$ ...
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0answers
26 views

Maximising an area

I was wondering if someone could possibly explain this question: "A stadium should be oblong on plan with straight sides of length h and semi-circular arcs of radius r at either end. The facade must ...
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0answers
11 views

An equality between maximums of two logdet expressions

I have the following question. Let $K$ be a positive-definite $N\times N$ real-valued matrix (I'll denote this by $0\prec K$ and will subsequently assume all matrices are $N\times N$ and real-valued) ...
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1answer
27 views

Application Farkas Lemma

Let $A$ be a $m \times n$ matrix and $C$ a $k \times n$ matrix. Let $b \in \mathbb{R}^m$ and $d \in \mathbb{R}^k$. Show that exactly one of the following holds: a) There exists an $x \in ...
2
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2answers
48 views

What is the interpretation of the following optimization problem?

Suppose we have $N$ variables $x_1,\ldots,x_N$. Let $\mathbf{A}$ a $M \times N$ matrix, and $\mathbf{b}$ a $M \times 1$ vector. I have the following minimization problem: \begin{array}{rl} \min ...
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1answer
20 views

Subgradient Method Example [on hold]

can anyone show me an example(function) of subgradient method(unconstrained)? subgradient algorithm https://inst.eecs.berkeley.edu/~ee227a/fa10/login/l_cvx_alg.html ...
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0answers
5 views

How to distinguish different states?

The linear model for sparse coding is y = Ax, where x is a sparse vector (m*1), A is the dictionary matrix (n*m), and y is the observation vector (n*1). If ...
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1answer
51 views

Is $\det(U_1\Lambda_1 U_1^t +U_2\Lambda_2 U_2^t +I)\le \det(\Lambda_1 +\Lambda_2 +I)$ correct?

I want to simplify or find an upper bound for the determinant $|K_1+K_2+I|$ where $I$ is identity matrix, $K_1$ and $K_2$ are positive semi-definite matrices of size $n$ and thus can be written as ...
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0answers
11 views

How to solve this optimization problem (Interpolation on trigram, bigram, and unigram for language model)?

I am a newbie in optimization and learn about the language model in NLP. I am studying the basic interpolation method to estimate the probability of the current word given the last 2 words, $P(w_i | ...
2
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0answers
35 views

Second-order Quadratic Constraint

I would like to solve the following optimization problem using the gradient ascend method: \begin{array}{ll} \text{maximize}_{\theta} & \theta^TQ_1\theta + b_1^T\theta\\ \text{subject to} & ...
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0answers
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How do we derive efficiency from robustness in the virtual ant solution to the traveling salesman problem?

Using virtual ants/swarm intelligence to solve the Traveling Salesman Problem is an example of using a robust system to solve an efficiency problem. We normally think of robustness and efficiency as ...
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135 views

Linear - Quadratic optimization for system of objectives

I have two distinct data sets, $\{x^{\mu},J^{\mu}\}$, $\mu=1,\ldots,n$ and $\{x^{\nu},V^{\nu}\}$, $\nu=1,\ldots,m$ that also include uncertainties $\delta J^{\mu}$ and $\delta V^{\nu}$. In these I fit ...
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2answers
31 views

Maximum perimeter for triangle inscribed in circle

How to prove that isosceles triangle has maximum perimeter from all trangles inscribed in circle? I found that from all isosceles trinagles - equilateral has maximum perimeter: Maximum perimeter of ...
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1answer
35 views

Finding the largest box inscribed in the ellipsoid

Among all rectangular boxes inscribed in the ellipsoid: $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$$ How to find the one with the largest volume?
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0answers
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Optimization on Stiefel Manifold

$$\text{Find}~~U, V$$ $$\text{to maximize}~~f(U,V)=\text{tr}(U^TAVN)$$ $$\text{subject}~~U^TU=I_p,V^TV=I_p$$ where $N=\text{diag}(\mu_1,\cdots,\mu_p)$ with $\mu_1>\mu_2>\cdots>\mu_p>0$. I ...
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3answers
31 views

Constrainted optimization involving logarithms

The problem is to minimize $ f(x_1, x_2 ,x_3, x_4):= - \Big[ \log ({\frac{1}{4} + x_1}) + \log ({\frac{1}{2} + x_2})+ \log ({\frac{1}{5} + x_3})+ \log ({\frac{3}{4} + x_4}) \big]$ such that ...
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0answers
12 views

Deriving Dual Averaging from (Sub)gradient Descent

Here the presenter tries to derive a simple Dual Averaging from (sub)gradient descent. I have a little problems understanding the steps. (Sub)gradient descent: Loop through: $$ x_{k+1} = x_k - t_k ...
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0answers
38 views

First fundamental theorem of calculus for line integrals [on hold]

Please, could someone look at this tricky question? Find the work done by force $F(x,y)=(3y^2+2) \hat i+16x \hat j$ in moving a particle from $(-1, 0)$ to $(1,0)$ along the upper half of the ellipse ...
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6answers
476 views

Finding the largest triangle inscribed in the unit circle

Among all triangles inscribed in the unit circle, how can the one with the largest area be found?
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1answer
34 views

Hessian-Matrix positive definite $\iff$ $a$ local minimum?

It is commonly known that if $f$ is twice differentiable, $\nabla f(a) = 0$ and $H_f(a)$ positive definite, $a$ is a local minimum. So, in short: $H_f(a)$ positive definite $ \implies $ $a$ local ...
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1answer
30 views

Lagrange primal and duality properties [on hold]

max Lprimal(x,lamb,mu) >= f(x*) + lamb g(x*) + mu h(x*) s.t. lamb >= 0 min Ldual(lamb,mu) = min max Lprimal(x,lamb,mu) What is the purpose and properties of the lagrange dual function, why do I need ...
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1answer
29 views

Dynamic programming approach for multidimensional problem

I use a dynamic programming approach to optimize the behaviour of individuals playing a game.I have one strategy matrix that describes the behaviour of individuals in situation 1, which depends on ...
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0answers
14 views

Functional Minimization of Exponential Decay

I would like to find a function $f$ that minimizes the functional: $$\ln(f(x))f(x)-\frac1x$$ over some range of $x > 0$. Is this a good application for functional calculus and the Euler-Lagrange ...
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0answers
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Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= ...
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1answer
45 views

Why does the functional have a local minimum at $0$?

Definition: Let $J: A \to \mathbb{R}$ be a functional , where $A \subset V$ and $(V, ||\cdot||)$ a linear space with norm. Let $y_0 \in A$ and $h \in V$ such that $y_0+ \epsilon h \in A $ for ...
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1answer
38 views

Is inequality $tr(A^{-1^T} B) tr(A^T B^{-1}) \leq constant$ correct?

I have the following optimization problem \begin{align} \min_{A} &tr(A^{-1^T} B)\cr \text{subject to} &x^T A x > 0 \cr & A_{ii}=1 \end{align} where $A$ and $B$ are some positive ...
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0answers
24 views

Maximizing the following function

I need to find values of $k_1$, $k_2$ and $k_3$ that maximize $C^{m_1}_{mm_1} \cdot C^{m_2}_{k_1-mm_1} \cdot C^{n_1}_{nn_1} \cdot C^{n_2}_{k_2-nn_1} \cdot C^{p_1}_{pp_1} \cdot C^{p_2}_{k_3-pp_1}$ ...
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0answers
12 views

Matching student-company at a fair (A variation of The Marriage Problem)

This problem is connected to the famous http://en.wikipedia.org/wiki/Stable_marriage_problem#Algorithm We have $s$ students and $c$ companies, where $s<c$. (Roughly speaking, $c \approx 20$ and $s ...
2
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1answer
40 views

Can critical point that $f''$ has different sign in its every neighborhood be a local extreme point?

Suppose that $f$ is a second order derivable function on $[0,1)$, and $f'(0)=0$. It is true that: If there exits $\delta>0$ such that $f''(x)\geq0$ for all $x\in[0,\delta)$, then $0$ is a local ...
0
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1answer
32 views

Representing a series of Matrix inner product with a single matrix product.

I have a set of constraints in my optimization problem, constraints in the form , $\langle A, e_i e_j^T \rangle = r_{ij} ,\forall i,j \epsilon S$, where $A$ is an $n*n$ semidefinite and symmetric ...
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1answer
81 views

How to minimize the expectation?

Given random variables $X_0, X_1, \ldots, X_n$ with finite expectations $m_0, m_1, \ldots, m_n$ I want to prove that the numbers $a_i = \frac{\det \Lambda_{i0}}{{\det \Lambda_{00}}}$ minimise the ...
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0answers
21 views

Solving the quadratic optimization problem with quadratic inequality constraint

I have a quadratic optimization problem which which both objective function and constraint are convex. As the problem is very big, I used decomposition technique and divide the problem to smaller ones ...