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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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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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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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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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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29 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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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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38 views

How to stop iteration in inverse problem using nonlinear least square problem?

I am having a real trouble with stopping criterion in iteration of Generalized Nonlinear Least Square. My problem is that I do not know exactly how to stop my iteration. First, I will give a short ...
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29 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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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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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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40 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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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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35 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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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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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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Integer Linear Programming, an optimization case. [closed]

A company has allocated one million dollars for the purchase of automobiles. The company has to choose between the manufacturer Alfa, which offers cars of a cylinder capacity 1800 at a cost of ...
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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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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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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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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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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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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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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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Linear Programming - Tableau Condition

The following tableau corresponds to an iteration of the simplex method: ...
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39 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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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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59 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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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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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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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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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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25 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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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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Maximum Likelihood Estimation for function with several variables

I have a function in four variables and I need to find out the variable values where the function will have maximum value. $f_n = (1-v1\cdot v2\cdot v3\cdot v4)\cdot (1-v1\cdot v2\cdot v4)\cdot ...
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Finding the source for minimizer of a functional for all $C^2$-curve $x(t_0)=x_0$ and $x(t_1)=x_1$

I am trying to find where this problem comes from and its corresponding proof for my students, but I cannot find the source anywhere. If anyone can find the source of this, or has any ideas where I ...
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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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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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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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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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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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Quasiconcave condition for a power function

Let $f(x, y)= (ax^2+by^2)^n$ where $a, b, n$ are positive, $x, y\in \mathbb{R}$. What is the condition of $n$ so that $f(x, y)$ is a quasiconcave, and concave function? My idea is only calculate ...
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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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virtual queue update

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, the ...
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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, ...