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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Consider the problem minimize $f(x)= x^4 −1. $

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
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3answers
32 views

minimize expression

How can the following expression be minimized wrt w: $$ \frac{w^T D w}{w^T S w}, $$ where $w \in \mathbf{R}^n$, $D \in \mathbf{R}^{n \times n}$ symmetric, and $S \in \mathbf{R}^{n \times n}$ ...
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1answer
42 views

Minimize the Frobenius norm of the difference of two matrices with respect to matrix: $\underset{B} {\mathrm{argmin}} \left\| A- B \right\|_F$

The following question is similar to this one, but I think that it is not straightforward to move from one to the other, so please take a look. Otherwise, please let me know and I will delete it. ...
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1answer
25 views

How is called this an optimization problem of this kind, or which techniques could I use to solve it?

I have an optimization problem which is a multivariable problem(34 variables), I need to find the minimum cost but my solution must be only concerning to the value of 3 variables out of the 34; the ...
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1answer
29 views

Sufficient condition for global maximum of strictly quasi-concave functions (unconstrained)?

Suppose $f(x)$ is a function from $\mathbb{R}$ to $\mathbb{R}$ and $f$ is strictly quasi-concave. If $x^*$ is a point such that $f'(x^*)=0$, then can we say that $x^*$ is a global maximum of this ...
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12 views

Are there any standard methods to solve a linear objective with nonconvex constraints?

I see that nonlinear programming entails nonlinear objectives with convex or linear constraints. Is there any theory/method to solve linear objective with nonconvex constraints and some convex ...
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1answer
18 views

Multivariable optimization - how to parametrize a boundary?

A metal plate has the shape of the region $x^2 + y^2 \leq 1$. The plate is heated so that the temperature at any point $(x,y)$ on it is indicated by $T(x,y) = 2x^2 + y^2 - y + 3$. Find the ...
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20 views

Differentiability of Moreau-Yosida Regularization? [duplicate]

I'm looking for a proof of the differentiability of the Moreau-Yosida regularization of a proper closed convex function $f(y)$ defined on an n-dimensional Banach space $Y$. namely the function is ...
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1answer
41 views

Proving convexity of a function whose Hessian is positive semidefinite over a convex set

C is a convex set in R^n and f:R^n --> R is twice continuously differentiable over C. The Hessian of f is positive semidefinite over C, and I want to show that f is therefore a convex function. I ...
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25 views

how to prove this sparse coding equation

How can I prove the following? $\sum_i \frac{1}{2} \|\mathbf{x}_i - D\mathbf{\alpha_i}\|^2 = \frac{1}{2}Tr(D^TDA_t) - Tr(D^TB_t)$ where, $A_t = \sum_{i=1}^T \mathbf{\alpha}_i\mathbf{\alpha}_i^T\\ ...
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2answers
28 views

Curiosity - maximising a product with a constraint

I have integers greater than 4, for instance $i_1$, $i_2$, $i_3$, ..., $i_n$. We have to change the greatest of these integers (for instance $i_1$ if they are ranked by descending order) by adding to ...
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36 views

solve the equation $c_1+c_2 e^{c_3 x}+e^{c_4 x}-e^{c_5 x}=0$

How can I solve the equation $$c_1+c_2 e^{c_3 x}+e^{c_4 x}-e^{c_5 x}=0,$$ where $c_1,\ldots,c_5$ are real numbers? I encountered this equation when I was solving a maximization problem. i can say only ...
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1answer
31 views

Second-price sealed-bid auction uniformly independent with unknown value

a disclaimer before the question: this is a homework problem. I just want some help/push in the right direction, I'm kind of stuck! The problem is as follows: In a second-price sealed-bid auction for ...
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1answer
44 views

minimum value of $x^2+y^2+z^2$ subject to $ax+by+cz=1$

If $ax+by+cz=1$, what is the minimum value of $x^2+y^2+z^2$ It is obvious that we can do Lagrangian multiplier,$W=x^2+y^2+z^2-\lambda (ax+by+cz-1)$
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100 views

Show that every extreme point in Q is either an extreme point of P or a convex combination of two adjacent extreme points of P

P is a bounded polyhedron in $\mathbb{R}^n$, $a$ a vector in $\mathbb{R}^n$, and $b$ some scalar. Define $$Q = {x \in P | a'x = b}$$. Show that every extreme point in Q is either an extreme point of P ...
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1answer
42 views

How to prove local minima are global?

I have the function $f(x,y) = (x^2 - 4)^2 + y^2,$ which has two local minima at $(2,0)$ and $(-2,0).$ How can I prove that these are global minima?
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30 views

Derivation of Steepest Descent Direction used in Line Search Methods

In the numerical optimization text I am reading, the Steepest Descent Direction was derived by considering $$ \min_{||p||_2\leq 1} p^T\nabla f(x_k) $$ This resulted in $$ p_k=-\frac{\nabla ...
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1answer
11 views

solve for the max of the sum of two points on a function a given distance apart?

I just thought of this concept and am not very experienced in math, so I'm assuming there's an easy solution I'm overlooking. For a given function y = f(x), how can one find the maximum value for the ...
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24 views

Extremal condition for series expansion coefficients

I want to maximize a coefficient in a series expansion, so the situation is the following. $f \in C^{\infty}$ and $f: \mathbb{R} \times \mathbb{R} \times [0,2 \pi] \rightarrow \mathbb{C}$. Now, we ...
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1answer
33 views

Maximum Likelihood Question

The aim is to find the maximum likelihood estimator for theta. $f(x)$ is given and we can assume that $1\le x\le-1$. I have completed the steps seen in the image, however I am having difficulty ...
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0answers
36 views

How to solve non-linear optimization problem with division

I am trying to solve the below mentioned optimization function. Here W_ck and W_lk have the same range but their positioning is such that for one calculation in k domain, one decision variable is in ...
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1answer
33 views

Prove that f has at least one global minimizer

$f: \mathbb{R}^n \to \mathbb{R}$ is a continuous function such that $\displaystyle\lim_{\|x\| \to \infty} f(x) = \infty$ On a side note: how can a function have more than one global minimizer? Is a ...
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1answer
45 views

Consider the problem minimize $f(x_1,x_2) = (x_2 −x_1^2)(x_2 −2x_1^2)$

(i) Show that the first- and second-order necessary conditions for optimality are satisfied at $(0,0)^T$. (ii) Show that the origin is a local minimizer of f along any line passing through the origin ...
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2answers
36 views

Maximum Likelihood Estimation Question

I'm really struggling with this question. From my understanding in order to find the maximum likelihood estimator for theta, the function needs to be partially differentiated with respect to theta ...
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2answers
45 views

Max-min inequality

It is known that $\underset{x}{\max} \underset{y}{\min} f(x,y) \leq \underset{y}{\min} \underset{x}{\max} f(x,y)$ . When does equality hold in this expression?
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17 views

Converting a derivative constraint into an orthogonality constraint

Let's say I'm trying to generate a quadratic curve in 3 dimensions, given two points it passes through, $\vec a$ and $\vec b$ in $\mathbb{R}^3$, and normals to the curve at those points, $\vec n_1$ ...
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0answers
15 views

How To Find a Set of Points Farthest Apart Within 3D Solid

I am trying to find out a method to solve the following problem: There are two parameters: 1) There is a solid 3D region plotted in a cartesian coordinate system. 2) There is a number of points that ...
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1answer
39 views

Know any “real life” optimization problems? (Constructing Functions)

Does anyone know "real world" optimization problems? The ones that relate to maximizing area and volume seem a bit contrived. For example, remember this old problem? An orchard has 800 orange ...
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9 views

Optimization of a set-based invariant for a single element case

As I understand it, various algebras have useful identities. For example, in boolean algebra, !a & !b is equivalent to ...
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8 views

Solution of Q*H=D via conjugate gradient?

I want to solve Q*H=D for Q given H and D via conjugate method(CG. Here Q, H, D all are matrices of suitable size. Can somebody help me out. Note that you can not multiple by inv(H) on the right and ...
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What can be said about a measure with given marginal measures

Let $(X,\mathcal F_X,\mu_X)$, $(Y,\mathcal F_Y,\mu_Y)$ be two measure spaces. Let $\mu$ be a measure on $\bigl(X\times Y, \sigma(\mathcal F_X \times \mathcal F_Y)\bigr)$ such that for each $A \in ...
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12 views

Integer linear programming problem.

A container has a capacity of 18 cubic meters. The container is designed to carry two types of goods: the goods of type A and the goods of type B. A is delivered in packaged units that occupy 3 cubic ...
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1answer
22 views

Maximum Value - Analytic function

I am having a hard time figuring out where to start and what results to use to address the following question: Suppose $f(z)$ is analytic in the unit disc $D=\{z:|z|<1\}$ and continuous in the ...
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2answers
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Can you help with this calculus problem? Optimization [closed]

Consider a rectangle that is inscribed with its base on the x-axis and its upper corners on the parabola $y=C−x^2$, with $C>0$. What are the width and height that maximize the area of this ...
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1answer
26 views

Feasible Condition with a single constraint

A linear program with a single constraint minimize $z = c_{1}x_{1} + c_{2}x_{2} +· · ·+c_{n}x_{n}$ subject to $a_{1}x_{1} + a_{2}x_{2} +· · ·+a_{n}x_{n} ≤ b$, $x_{1}, x_{2}, . . . , x_{n} ≥ 0.$ (a) ...
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1answer
29 views

Optimising using Hessian matrix

I am bit perplexed in optimisation problem if the principal minor is zero. If the principal minor is zero does it mean that the Hessian matrix is always indefinite and the point of extremum will refer ...
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13 views

A Step in the proof of Rate of Convergence of Steepest Descent in Nocedal's Numerical Optimization

I am not sure how to derive it properly. I tried to expand the LHS first: $f(x_{k}- \alpha \bigtriangledown f_{k}) = \frac{1}{2} {x_{k}}^{T} Q x_{k} -\frac{1}{2} \alpha {f_{k}}^{T} Q x_{k} - ...
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1answer
41 views

Set of optimal solutions for a linear programs

Consider the linear program: minimize $z = x_{1} - x_{2}$, $x_{1}, x_{2}\geq 0$ subject to: $-x_{1} + x_{2}\leq 1$ , $x_{1} - 2x_{2}\leq 2$ Derive an ...
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50 views

Linear Programming - Tableau Condition

The following tableau corresponds to an iteration of the simplex method: ...
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1answer
62 views

How to maximize area of two circles inside a rectangle without overlapping?

Two circles have to be drawn inside a rectangle of dimensions $W\times H$ such that the area of both circles is to be as large as possible without overlapping. Let the radii of the circles be $r_1$ ...
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4answers
54 views

optimization of coefficients with constant sum of inverses

Does anybody knows if there is an easy solution to the following problem: Given $A = [a_1, a_2, ... a_n]$ and K, find B = $[b_1, b_2,...b_n]$ that minimizes $AB^T$ such that ...
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1answer
65 views

Why geometric median cannot be solved analytically

$\DeclareMathOperator*{\argmin}{argmin}$ For a given set of $m$ points $x_1,...,x_m$ with each $x_i\in \mathbb{R}^n$, the geometric median (or the weber point) is defined as $$\argmin\limits_{y \in ...
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1answer
36 views

Find the max volume using polynomials with the sum of the height and perimeter less than 100cm

I have to find out which shape of packaging for a fragile object has the most volume to fit the object and styrofoam packing. The sum of the height and the perimeter must be less than 100cm. There is ...
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21 views

Minimum of an Entropy based function

This question is a small part of a bigger problem I am working on. Let $h(p)$ be the binary entropy function. That is, for $p \in (0,1)$ $$h(p) = -p\log_2(p) - (1-p)\log_2(1-p)$$ Define the ...
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Does this formula take constant value?

Now, $x_i, \xi, f \in R^n(i= 1, 2, \cdots , k)$, and \begin{align} \sum_{i=1}^k x_ix_i^T\xi=f \end{align} holds. If the above equation is solvable about $\xi$, the value of $f^T\xi$ doesn't depend on ...
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0answers
21 views

A Difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$. (Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...
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12 views

How to obtain closed form solution to the constrained optimization problem?

Suppose the following minimization problem: $$ N^*(\lambda)=\min_{X\in\mathbb{R}^8}\left\|D\left(A\cdot X-b\right)\right\|^2_2 \\ s.t. C_\lambda X= r_\lambda, $$ where $X\in \mathbb{R}^{8\times ...
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9 views

Convergence results for incremental generalized gradient methods

Are there convergence results of incremental and stochastic subgradient / generalized gradient methods for locally Lipschitz functions that are not necessarily convex? I am mainly interested in ...
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1answer
28 views

Show that a matrix A may have all leading principal minors greater or equal to zero, yet not be positive semi-definite.

Title says it all, but I'll rephrase it to be clear. A is an $n\times n$ matrix whose leading principal minors are all greater than or equal to zero. A leading principal minor is the determinant ...
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53 views

How to find if an integral is possible to compute: Failing to solve integral for quadratic functional

I am trying to solve the below integral, and no computational method seems to be capable of solving this, nor can I do it by hand. Any ideas? $$\int_{t_0}^{t_1}[a(t)((2\dot{x^*}\dot{\eta} + ...