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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Convexity Proof with constraints on the gradient

Consider a minimization problem $(P)$ : minimize $f(x)$ subject to $\delta_C(x) \leq 0$ Now assume that $\emptyset \neq C \subset \mathbb{R}^n$ is convex and let $f: \mathbb{R}^n \to \mathbb{R}$ be ...
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Would I use KKT conditions to solve this optimiztion problem?

my problem (P) is: $$(P) \space \space \text{min} \space x_1x_2$$ $$\text{s.t.} \space x_1-x_2-2 \leq 0$$ $$x_2 \leq 0$$ Prove that $x^* = (1,-1)$ is a strict local minimizer. ...
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optimization problem resolution

I need to get the value of k that minimizes : $\frac{k}{\sqrt{k^2+\left( \omega-\omega_{c}\right)^2}}$ under the constraints : $k > 0$ and $\omega \ne \omega_{c} $. Thank you.
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A minimum value question?

This is what the original question was: The minimum value of the expression $\sin A + \sin B + \sin C$, where $A$,$B$ and $C$ are real numbers satisfying $A+B+C=\pi$ is (A) positive (B) zero (C) ...
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How to prove the existence of a minimum of a quadratic function of two variables?

I am given function $$ f(x,y)=Ax^2+2Bxy+Cy^2+2Dx+2Ey+F,\quad\text{where }A>0\text{ and }B^2<AC . $$ Prove that a point $(a,b)$ exists which $f$ has a minimum. I figured out that there is ...
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329 views

Sub-gradient of the “$\ell_0$ norm”

I am trying to characterize the sub-gradient of l0-norm ($f(x) = ||x||_0=\sum_{i=1}^n 1\{{x_i \neq 0}\}$). At first, I thought l0-norm is a convex non-smooth function since it satisfies the triangle ...
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Question about Lagrange multiplier and maximum point

Find the maximum of $\log{x}+\log{y}+3\log{z}$ on portion of the sphere $x^2 + y^2 + z^2 =5r^2$ where $x,y,z>o $ I found that maximum is $5\log{r} + 3\log{\sqrt{3}}$ at $(r,r,3\sqrt{3})$ And ...
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244 views

How can I find minimum volume using Lagrange multipliers?

What is the minimum volume bounded by the planes $x=0, y=0, z=0$ and a plane which is tangent to the ellipsoid $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1$$ where $x,y,z>0$ I only ...
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42 views

expectation of a minimization problem

Dose the following inequality hold? $E_i[min_{y\in D} \ \ y^Tx_i] \leq min_{y\in D} \ E_i[y^Tx_i]$ Please help me to prove or disprove the above inequality.
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How to get this upper bound of this sum of squares?

Given n non-negative values. Their sum is k. $x_1+x_2+⋯+x_n=k$ Given the constraints $x_i \leq \sqrt{k}$ (thus, $n \geq \sqrt{k}$) Is it possible to prove that $x_1^2 + x_2^2 + ... + x_n^2 \leq ...
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Problem about length and width of a running facility

Jacaranda Secondary College is planning to develop a $400$ metre running track facility in an unused area of the college. The rectangular site available is $100$ metres wide and $180$ metres long. The ...
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539 views

What minimizes the Chebyshev Distance?

For an arbitrary number of dimensions, I know that the mean minimizes the distance using the L2 norm and that the geometric median minimizes the distance function using the L1 norm (though I have yet ...
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178 views

Wrong ILP solution with LPSolve (simple example)

I added the following example into LPSolve and found a strange issue. I don't want S1 and S2 to overlap within certain margins. ...
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130 views

Mminimize the integral and obtaining the constants $a$ and $b$

Determine the constants $a$ and $b$ for the integral $$ \int\limits _{0}^{1}(ax+b-f(x))^{2} dx$$ take the smallest possible value if $f(x)=(x^{2}+1)^{-1}$ thanks
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289 views

Second Partial Derivative Test for a Three-Variable Function

Here is the function in question: $$f(x,y,z) = x^2 + x^2y + y^2z + z^2 - 4z$$ I need to find all critical points and use the second derivative test to determine if each one is a local minimum, ...
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solving euler-lagrange equation in constrained functional optimization

The problem to solve is the minimization of a functional of two functions, $F(y,z) = \int_a^b f(y,z)dx$ , subject to a constraint $g(y,z,y',z') = 0$. The augmented functional is then $L(y,z,y',z') = ...
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Question About Local minimum

I have this definition of a local minimum: We say that $u$ is a local minimum of $f$ is there exist a neighborhood $V$ of $u$ such that for all $v\in V$ $f(v)\geq f(u).$ So we say that $u$ is not a ...
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L_1 norm optimization as a sequence of linear optimizations?

Does someone know of numerical methods to approximately solve ${\bf x_0} = \min_{\bf x}\{ \left\|\bf Mx - b\right\|_1\}$ by using some sequence of linear optimizations? Links or ideas are both ...
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When $\min \max = \max \min$?

Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be compact sets. Consider a continuous function $f : X \times Y \rightarrow \mathbb{R}$. Say under which condition we have $$ \min_{x \in ...
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Solution techniques for optimization problems

I am very new to solving such optimization problems. Following is the problem, I need to know the various methods (preferably advanced machine learning techniques) that I can use to solve this. ...
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36 views

descent versus ascent direction

Suppose $f$ is a function from $\mathbb{R}^n \to \mathbb{R}$ If $\langle\nabla f(x), p \rangle$ =0, is $p$ an ascent or descent direction for $f$ at point $x$?
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Steepest-descent optimization method

I was wondering if the steepest descent method can find a global min/max or only local min/max? Thanks
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105 views

Topology and Differential Games

I'm a engineer who is making research on differential games in multiagent control. I was reading a tutorial on differential games and the author advised to get the required math background from the ...
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the continuity of argmin on convex funtion

Define $$x'=\text{argmin}_{x_1}f(x_1,\lambda),$$ where $f$ is a strictly convex function on $x_1$ and $\lambda$. I would like to ask if there is any theorem about the continuity of $x'$ w.r.t ...
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Integer Linear Programming

Without using a computer, I have to solve the following integer linear programming:$$\min \quad x_1+x_2+x_3$$ $$\operatorname{sub} ...
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Rank, nuclear norm and Frobenius norm of a matrix.

The nuclear norm, denoted $\|\cdot\|_*$ is a good surrogate for the $rank$ when minimizing problems like $$\label{pb1}\tag{1} \min_X rank(X) : AX = B $$ Here, we're trying to find an matrix X with low ...
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133 views

Maximum of the minimal distance of a set of points in an equilateral triangle

In this question, a closed triangle on a plane is a set of all points in its area and on its boundary, while an open triangle excludes its boundary. Now, the problems: Let $T$ be an equilateral ...
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31 views

Is this a convex program?

I have a nonlinear optimization problem $\min \sum_{i=1}^n \sum_{j=1}^n y_{i,j}$ subject to $x_i- y_{i,j}x_j\leq 0$ $0\leq x_i\leq 1$ $y_{i,j}>0$ The question is whether this is a convex ...
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Optimization problem using calculus from Khan Academy

Consider the function $f(x)=9−x^2$ for $f(x)≥0$ only. Let $T(x)$ equal the area of the shaded isosceles trapezoid that has two vertices on the x-axis and two vertices on the graph of $f$, as ...
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How to plot feasible objective space of a Generic Multiobjective Optimization problem?

As you know, a generic Multiobjective optimization problem can be stated as follows: $min{\space}F(\bf{X})=[f_1(x),...,f_n(x)]$ $h_k(x)=0{\space\space\space} k=1,...,n_e$ ...
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How may times should I colour a colour palette to have distinct colours?

Suppose that we have a colour palette, i.e., an array of n elements, which needs to be coloured by distinct numbers. We are only allowed to use 0 or 1 to colour every elements in each colouring step. ...
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How do you minimize $\varphi(f) = \int_y \tilde{\varphi}(f(y),y) p_y(y) dy$?

I was trying to minimize the following equation (over choices of $f$): $$\varphi(f) = \int_y \tilde{\varphi}(f(y),y) p_y(y) dy$$ where $p_y(y)$ is nonnegative and $\tilde{\varphi}(f(y),y) = ...
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Optimization, prove the following equation…

Proving the equation in part (a) is difficult, it may require using (t=d/s). However in part (b) my answer i got was 50 gallons per hour. Can someone please provide help with part(a).
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Probabilistic modelling of PSO algorithm's first iteration, mean of distribution of minimum

The particle swarm optimization algorithm (PSO) consists of a set of $I$ particles, each having a velocity $v_i$ and position $x_i$. The algorithm keeps track of the best encountered position for each ...
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396 views

Prove a property of primal-dual problems

When I was studying the computation aspects of quantile regression, I consulted some linear programming book and found an interesting property as follows: If the primal problem have unbounded ...
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192 views

Fan speed algorithm

I'm a programmer an I think my problem related to mathematics! I want when CPU have a static percentage of load (for example $10\%$) fan also have static rpm (Rotations per minute). But for now I have ...
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How do distribute exams within three weeks?

In our program of study we have $N$ students and $M$ courses. I guess $M \approx 50$ and $N \approx 1000$. Each student takes at most $3$ courses in our department. At our university we struggle with ...
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Why is this set a subset of its polyhedral approximation - contradicting the gradient inequality?

Say we have a set $C:= \{y\in \mathbb{R}^n : g_i(y) \leq 0, \space i=1,...,m\}$ where $g_i : \mathbb{R}^n \to \mathbb{R}$ are convex and differentiable functions, then we have $\tilde C : = \{y: ...
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Maximum value, minimum value of $|x^2 +2x -3 | + 1.5 \ln x $

Find the maximum and minimum value of function: $$F(x)= |x^2 +2x -3 | + 1.5 \ln x $$ over the interval: $$[\frac{1}{2},4]$$ $(21 +3\ln 2, -1.5\ln 2)$ $(21 + \ln 1.5,0)$ $(21 + 3\ln 2,0)$ $(21 +\ln ...
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401 views

Converting nonlinear program into linear program

Consider the following nonlinear optimization problem \begin{align} \min \quad c^Tx &+ f(d^Tx)\\ \text{s.t.} \quad Ax &\geq b\\ x &\geq 0 \end{align} where $$ f(y) = \begin{cases} -y+2 ...
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Maximize area in $1000\times 1000$ array using two non-overlapping squares centred at two points, p1 and p2.

Maximize area in $1000\times 1000$ array using two non-overlapping squares centered at two points, p1 and p2. The following conditions must be met: The area (square 1 area + square 2 area) should be ...
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What is a Single Objective Optimization problem?

I can't find any definition of this problem on the Internet. Could you help me by providing some definition?
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Circle is inscribed in isosceles triangle with area $2$. Find angles of triangle for which radius of circle is maximal.

Circle is inscribed in isosceles triangle with area $2$. Find angles of triangle for which radius of circle is maximal. I have $\displaystyle r=\frac{4}{\frac{4}{\sqrt{\sin ...
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Constraint minimization of sum of Non-symmetric matrices

I am trying to find closed form solution to following problem \begin{equation} \begin{array}{c} \text{min} \hspace{4mm} \big(\lambda_1\left( \mathbf{y}^T V^{(1)}\mathbf{x} \right)^2 + ...
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Analyze the variation of this function $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$ w.r.t. $x$

Please, I need to analyse the variation of the following function w.r.t. $x$ : $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$, where $E_1[a+b (x-1)]$ is the exponential integral, $b>a$, $a>0$, ...
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Conversion into linear program

I have an optimisation problem with decision variables that are multiplied with another (a weighted average is calculated). I'd like to convert it into a linear program. I found this link that ...
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Does type of a critical point depend on domain?

Suppose we have a function $f: D \to \mathbb{R}$, for a domain $D$, and $X \in D$ is a saddle point of $f$. Is it possible that if we constrain $f$ to a new domain $C$, where $C \subset D$, the same ...
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Prove that optimal solution is an extreme point in LPP.

While proving this I have proved that Optimal solution cannot lie inside the feasible set and that each supporting hyperplane to a set bounded from below (which is the case as in LPP we can always set ...
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An Optimal Value of a Diagonal Matrix $\Xi$ in $ H = U \Xi$

We have access to very accurate estimates of matrices $H$ and $U$ (both are $n \times k$, $n > k$) such that the following relationship holds $$ H = U \Xi$$ where $\Xi$ is a $k \times k$ diagonal ...
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Maximizing the number of zero entries in a linear combination of matrices

I was wondering if there exists an algorithmic way of solving the following problem. Let's say you have a bunch of square $N\times N$ matrices (call them $M_i$), and you want to form a linear ...