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 find maximum value of trig function?

How to find maximum value of this: $$y = 5\sin x - 12\cos x$$ And I am more intrested in solving process, rather than answer. I know the answer. I am familiar with derivatives, not so good, but as I ...
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Terminal Condition in Pontryagin Maximization

I'm doing a time dependent maximization problem using Pontryagin. Now the necessary terminal condition for a solution is only sufficient if my terminal function is concave. If my terminal function is ...
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29 views

Convert a nonconvex function to convex function

I have a image $I: \Omega \to \Bbb R$. It is separated into 2 non-overlapping region: $D$ and $\Omega \setminus D$ Each point $x$ in the image $I$, the $\phi$ function is defined as: $$\phi(x)= ...
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Proving the existence of $b$ such that $\prod_{k=1}^n(1-\cos(a_k-b))=\frac{1}{2^n}$

Let $n>0$ and $a_1,\ldots,a_n\in \mathbb R$. Prove there is some $b$ such that $\prod_{k=1}^n(1-\cos(a_k-b))=\frac{1}{2^n}$ This is motivated by this question Finding a point on the unit ...
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Justifying the “Dual feasibility”, one of the Karush-Kuhn-Tucker conditions

I am having difficulty of interpreting the KKT conditions in a general setting where we have $M$ equality and $N$ inequality constraints defined as: Minimize $f(x)$ subject to $g_i(x) \leq 0 , h_j(x) ...
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Linear Optimization - functions in constraints?

This is a real-world linear optimization problem. The model tries to minimize cost of hiring employees over a two month period, while providing a certain amount of service. Employees have a salary ...
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18 views

Find the signs of elements in a list such that their sum is equal to zero

I have a set $X = \{x_1, x_2, \dots x_N\} \in [0;1]^N$ containing $N$ elements, initially all positive. My goal is to find a vector of signs $S = \{s_1, s_2, \dots s_N\} \in \{-1; 1\}^N$ such that: ...
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Finding the minimum value of a function

Find the least value of $f(x)=3^{-x+1} + e^{-x-1}$. I tried to use the maxima/minima concept but it was of no use. Please help.
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Analysis of iterative optimization methods using lyapunov analysis

In analysis of iterative methods, is it possible that we have to use two time-lagged version of the time-varying system to analyze its convergence? (that is, we construct the evolution of x^k, ...
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39 views

Has extremum or not?

I'm learning calculus and I have to do with functions $x^2\sin(\frac{1}{x})$ where x!=0 and 0 when x=0 and $x^3\sin(\frac{1}{x})$ where x!=0 and 0 when x=0 If I computed this well, both of them have ...
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Mixed Integer Linear Program (MILP) question

I am trying to solve an MILP problem. I was wondering if Branch and Cut/Branch and Bound methods find optimal solution or not? Isn't the complexity exponential? Are there heuristic solvers available? ...
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Quite rare kind of proof of convexity for a quadratic function!

Excuse me all of you in advance. I got this problem as an assignment but I am not really good doing proofs! If $f(x)=\frac12x^TQx+b^Tx+a$ is quadratic in $n$ variables, where $Q$ is symmetric. Show ...
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1answer
23 views

Least sum of products of powers

Numbers from the set $\{2^1, 2^2, ..., 2^{10}\}$ are somehow permuted and paired with numbers from the set $\{3^1, 3^2, ..., 3^{10}\}$. Numbers in each pair are multiplied and the products are summed. ...
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Determine min and max of the following expressions

Let $a_1,\; ...\;, a_n$ and $b_1,\; ...\;, b_n\in \mathbb R$ be positive real numbers. Find $$ max \;(a_1x_1 + a_2x_2 + ... + a_nx_n) $$ and $$ min \;(a_1x_1 + a_2x_2 + ... + a_nx_n) $$ over $x_1, ...
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42 views

How to prove $f(x_1,\ldots,x_n) = \sum x_i\ln x_i - (\sum x_i )\ln(\sum x_i)$ is convex on R++ [duplicate]

How can i prove $f(x_1,\ldots,x_n) = \sum x_i\ln x_i - (\sum x_i )\ln(\sum x_i)$ is convex on R++
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1answer
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What is the approach to solve simple constrained optimization when first order condition $\nabla f = 0$ yields solution outside of domain

I wish to solve the problem min $ f(x_1, x_2) = x_1^2 - x_1 + x_2 + x_1x_2$ subj $x_1\ge 0, x_2 \ge 0$ We find $\nabla f = [2x_1 - 1 + x_2, 1+x_1] = 0$ yields $x_1 = -1, x_2 = 3$ which is outside ...
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Optimizing the trace of a matrix product

I have a problem where I have a NxT matrix P (lets just assume full rank for now, where N>>T) and a TxN inclusion matrix S. Each column of S must contain exactly one 1 and the rest 0's i.e. 1_T*S = 1, ...
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67 views

The best strategy to increase StackExchange Reputation [closed]

I do not have a lot of background in game theory, but I am curious how would one formally pose the title problem and mathematically describe possible strategies. Are the problems of this type best ...
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Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$

I have been stuck at this problem for a while :( Given $\mathbf{A}\in\mathbb{S}^{p\times p}, \mathbf{A}\ge 0,\mathbf{A} \text{ symmetric}, \mathbf{b}\in\mathbb{R}^n,\mathbf{c}_i\in\mathbb{R}^p\forall ...
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1answer
22 views

Is this reducible to a standard optimization problem?

There are $N$ agents who needs to be allocated $K$ discrete resources. There is a bottleneck threshold utility $R$ per agent. The $i$th agent has utility $r_{ij}$ if he is allocated $j$th resource. ...
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Max-min of a function on closed, bounded interval using EVT

I'm just having little bit of difficulty with the following question: Find the local maxima and minima of $f : [0, 1] \rightarrow \mathbb{R}$ defined by $$ f(x)=x^4(1-x)^6 $$ So we know the ...
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Multiple optimal solutions / LP

In the optimal primal simplex tableau, if we have a non-basic variable with a reduced cost of zero, can we say for sure the primal has multiple optimal solutions? Or can the same thing also happen ...
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Assignment problem with multiple types, capacities and costs

I am trying to solve an optimization problem (variation of assignment problem). I'm stuck with how to represent this problem (as an LP or graph based). If it's formulated as a LP, I'm unsure of how to ...
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How to compute primal variable based on dual variables and their multipliers

I edited this question based on information I got from comments. Assume we have an optimization problem (primal problem). we solve it's dual using some kind of primal-dual interior point solver. So, ...
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tangent cone to the set

I'm supposed to solve this problem: Let us consider the set $M=\{(x, \sin{x}):x\in\mathbb{R}\}\cup\{\big(\cos(x)-1,x\big):x\in\mathbb{R}\}$ The question is to find the tangent cone to the set $M$ in ...
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20 views

Derivative vector and Hessian for maximization

I'm having some troubles regarding maximization of approximated utility. I want to use the Newton method, but in order to do so I need the derivative vector and the Hessian matrix (I will be ...
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26 views

Karush-Kuhn-Tucker conditions for non-linear optimalization

I have the following problem: solve the local conditions (KKT) and find ALL optimal solutions: $$\min f(x,y)$$ subject to $$g(x,y)\le 0$$ $$x\geq0, y\in\mathbb{R}$$ I have some questions to this ...
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Saturation Curve

I have a equation which is x/(x+40). I'm trying to find a point indicated in the graph. As you can see i drew 2 lines, one tangent to the region which it saturates, the other were it has max growth. ...
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Rounding a distribution to minimize loss

This question deals with the problem of choosing cutoff points such that rounding a random variable down to the nearest cutoff point doesn't lose "too much" of its mean. Formally: Let $y$ be a ...
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Upperbound for a linear algebraic ratio?

Consider ($n\times 1$)-column vector $\mathbf{p} = (p_i)_{i=1}^n$ with $p_i > 0$ and a symmetric ($n\times n$)-matrix $\mathbf{A} = [a_{ij}]$ with $a_{ii} = 0$ and $a_{ij} \in [0,1]$ for $i \neq ...
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1answer
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Minimizer of a quadratic form

Suppose I have a quadratic form of the form: $$q(x)=\frac{1}{2} x^T Q x$$ Now I want to find the minimum step length w.r.t the steepest descent. So I know the descent direction is $\nabla q(x)$. So ...
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1answer
91 views

Optimization of $f\left(x\right)=x^{2}\sin x^{3}$

Let $$f\left(x\right)=x^{2}\sin x^{3}$$ Set of critical points consists of isolated points. Set of critical points is compact $f(x)$ attains local extremum at any critical point ...
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How to divide items into nearly equal sized groups

I teach a class with about $N=290$ students who will be taking an exam next week. The exam consists of two sections ($A$ and $B$) each with 6 essay questions. Students must answer one essay question ...
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Proof that equation is non-convex function

I have a objective function as following $$E(\phi)=\int_{\Omega}(I(x)-m_1)^2H(\phi(x))dx+\int_{\Omega}(I(x)-m_2)^2(1-H(\phi(x)))dx+\int_{\Omega}|\nabla H(\phi(x)|dx$$ where $I$ is an image; $I: \Omega ...
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optimization on two “max” function

Anyone knows how to use lagrange multiplier (or KKT conditions) to minimize an objective function such as $L(\beta,\beta_0)=\sum_{i=1}^n[a_i(1-y_if(x_i))_++b_i(1+y_if(x_i))_+$] where $a_i$, $b_i$ ...
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Traveling salesman problem (TSP): what is the Relation with number of vertices and length of the found route?

I know that there are many algorithms (exact or approximate) which implement the traveling salesman problem. I would like to know the relation between the number of the vertices (i.e., the places to ...
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1answer
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A weird Calculus of Variations problem

I became stuck with the following Calculus of Variations problem. The problem is related with something called as the "Nadaraya-Watson" model in statistics. We have $N$ inputs ${x_n}$ and each of ...
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1answer
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What is the maximum value of the sum $\sum_{i=1}^L(\bar{x}-x_i)$, in this specific case.

Let $x_i$ be a positive real variable, with $i=1,2,...,K$. We denote by $\bar{x}$ the average value of the values $x_1, x_2,...,x_K$. Let $a=\min_i x_i$ and $b=\max_i x_i$, then $x_i \in [a,b]$. My ...
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Devising a likelihood method for estimating disease prevalence in hunted deer populations

I am attempting to find the maximum likelihood estimate for disease prevalence in trapped mice by using data on the probability of being trapped each year and the number of mice actually trapped that ...
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1answer
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Motivated by Level Sets, how can I show that minimizing this functional is equivalent to this PDE?

I would like to show, that minimizing the functional $$F(g)=\alpha\int_\Omega |\nabla g(x)|^2dx+\mu \int_\Omega (g(x)-f(x))^2dx $$ is equivalent to solving the differential euqation $$-\alpha\nabla ...
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Lagrange Multiplier with Inequality Constraints

To maximize $f(x, y)$ subject to $g(x, y) \le b$, we define a Lagrangian $$L(x, y, λ) = f(x, y)−λg(x, y).$$ Then the conditions are: $$Lx = 0,\ Ly = 0,\ \lambda(g(x, y) − b) = 0,\ g(x, y) \le b$$ ...
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ZELDA Guardian Puzzle Part II - Shortest Path (Unsolved for new rules)

This question is in relation to the following previously asked question: Twilight Zelda Guardian Puzzle : Shortest Path (UPDATE: ADDED RULES) A 1-step-less solution was uncovered, but under an ...
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1answer
20 views

Perfect matching problem

We have a random graph G = (V,E). Two players are playing a game in which they are alternately selecting edges of graph so that in every moment all the selected edges are forming a simple path (path ...
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1answer
49 views

matrix gradient

I found the gradient of an optimization problem as $$ b*I + \rho\big(-A+diag(A)+X-2diag(X)\big) = 0 $$ But my problem is, I want to find the equation for $X$. From the above equation, because of the ...
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Must Number of equality constraints and decision variables be equal?

Must Number of equality constraints and decision variables in an optimization problem be equal ? If not, how can I solve the equality constraint equations with a solver e.g. ...
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Find an example of critical point

Find an example have the following property: Let $\Omega $ be open in $\mathbb{R}^{n}$, $f, g : \Omega \rightarrow \mathbb {R}$ be $\mathcal{C}^{1}(\Omega)$ and $S=\begin{Bmatrix} ...
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find mean of matrices $A_i, A_j$ given $d_{A_{ji}}=\ln{\left|\left| A_{ji} \right|\right| \left|\left| A_{ji}^{-1} \right|\right|}$

Given a finite set $\mathbb{A}$ of $k$ like-shaped, square, non-singular matrices $A_i\in\mathbb{R}^{n\times n}$, let's define $A_{ji}=A_j A_i^{-1}$, then the distance of the two matrices $A_i, A_j$ ...
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Optimization of sum of logs

I have an optimization problem of the form $$\operatorname*{argmax}_{\mathbf{w}} \sum_i \log(1 + \mathbf{w} \cdot \mathbf{k_i})$$ given some set of vectors, $\mathbf{ \{k_i\} }$. I have tried both ...
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1answer
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Dealing with free variables in Linear Programming

I have a free variable in my formulation. In the objective function, this free variable has a cost, and another cost coefficient which is only incurred when the free variable is negative. I used the ...
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4answers
43 views

Monotonicity of $f(x)= \frac{ e^{\alpha x}-1}{\alpha x}$ where $\alpha \geq1 $ and $x >0$.

There is this function I encountered when I was solving a problem and I am trying to study its monotonicity. So the function is $f(x)= \frac{ e^{\alpha x}-1}{\alpha x}$ where $\alpha \geq1 $ and $x ...