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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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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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
226 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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33 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
266 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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1answer
738 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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2answers
47 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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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
181 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
50 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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288 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
146 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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115 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
22 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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1answer
45 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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1answer
163 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
77 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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121 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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3answers
423 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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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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1answer
102 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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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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1answer
46 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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69 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
55 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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1answer
144 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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402 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
72 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
153 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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155 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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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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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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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} + ...
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Algorithm to find the “optimal” path in a given graph

Assume that $G=(V,E)$ is an undirected connected graph and that $H: V \to \mathbb R$ is a function that assign at each vertex $v \in V$ its height $H(v)$. Think of the pair $(G,H)$ as an energy ...
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Spacing of fence posts with minimal distance to other fence posts

Definition 1: A "fence" is a set of "fence post positions", where each pair of adjacent positions has the same difference (the spacing), e.g. $\{1,2, 3, 4\}$. A fence is described by three values ...
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1answer
15 views

Max no. of piece in k cut

Suppose I have large piece of rectangular sheet. Cutting is allowed only vertically and horizontally. My approach is if no. of cut is even then max. no of piece is (n/2)*(n/2) if no of cut is odd ...
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157 views

Which shape optimize the perimeter for a given area?

I was wondering what is shape that maximize the perimeter for a given area? In fact I would like to know what would be the most optimized perimeter that I can include in a rectangle. See image ...
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1answer
113 views

What to do *rigorously* when the second derivative test is inconclusive?

How do you rigorously check if a point is a local minimum when the second derivative test is inconclusive? Does there exist a way to do this in general for arbitrary smooth (or analytic...) functions? ...
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1answer
303 views

Unique extrema of sum of monotonically increasing and decreasing functions on an interval

If I have two functions, f and g, defined on the interval [0, 1] with both f and g non negative (i.e. f(x), g(x) >= 0) f(x) is monotonically increasing, while g(x) is monotonically decreasing. and ...
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Critical points characterization of real function

Providing a real multi-variable function $f(\bar{x})$ twice differentiable with respect of all its variables. Looking for critical points is equivalent to solve $\nabla f = \vec{0}$. And to ...
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how to use linear programming for Heaviside Step function and L1 norm?

I want to find a hyperplane that can divide my sets of points into 2 groups that have nearly equal size. If the hyperplane is $w$, there is a scalar offset $b$. I have $N$ points that are ...
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Dynamic or virtual Queue

I am trying to formulate one equation. Let $Q(t)$ represent the contents of a single-server discrete time queueing system defined over integer time slots $t \in \{0, 1, 2, . . .\}$. Specifically, ...
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1answer
42 views

Finding the extremal curve satisfying a variable endpoint

Below is a question I am trying to solve, and my attempt. $\int_0^T \frac{\dot{x}^2}{t^3} \mathrm{d} t$, where $x(0)=1 $ and $x(T)$ lies on the curve Transversal condition: $$f-(\dot{c} ...
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About a step in the derive of Netwon Method

I do not understand a step in the derive of Newton method in my lecture notes: When it applies derivative on both sides of $q(x) = f(\bar{x}) + \bigtriangledown f(\bar{x})^{T}(x-\bar{x}) + ...
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1answer
95 views

Optimization: KKT conditions statement

I'm currently following this material Optimization Theory: Chapter 2 Theory of Constrained Optimization And I can't understand why the following statement is true, between the equations (2.9) and ...
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To minimize surface area of integer cuboid of ​​the known volume

There is a cuboid (a * b * c), (a, b, c ∈ N). S (Surface area of a cuboid) = 2 * (ab + bc + ca). V (Volume of a cuboid) = a * b * c = n. I need to minimize S, provided that I specified the volume ...
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Adjust previous optimal solution to new assignment problem

Suppose I have an assignment problem with $n$ workers and $n$ jobs and its optimal solution. Now another worker and another job comes along and we are given all new costs. Is there an efficient ...
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Kuhn-Tucker constraint qualification, overdetermined?

I have a question about the constraint qualification for KKT. As I've seen the theorem stated if $G(x^*)=(g_1(x^*),\dots,g_n(x^*))$ are the binding constraints at a local max $x^*$ then the jacobian ...
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1answer
83 views

Minimal disjoint chains covering graph vertex set

I'm looking for references on the following problem: Given a graph $G=(V,E)$, what is the minimum number of simple, disjoint paths that span all the vertices in $V$? i.e., let $P$ be the answer to ...