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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Search for sharp maximums of 4D surface

I need to find sharp local maximums of numerically defined 4D surface. I have a surface with lots of maximums. I already know how to find them all. Some of them look like this: wide extremum, others ...
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Extremum of a multidimensional quadratic function

I have the following function: $$ g(h) = h'\Sigma\Sigma'h-h'm-r, $$ where $h$ is a vector in $\mathbb{R}^M$, $\Sigma$ is a $M\times K$ matrix such that $\Sigma\Sigma'$ is positive definite and has ...
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How many methods could be used to solve this optimization problem with equality constraints?

I wonder whether there is a simplest method for this problem. The function to maximize is $F(x)$. $F(x)=\|Kx\|_2^2=x^TK^TKx$, where $K\in \mathbb{R}^{n\times d}$ and $x\in \mathbb{R}^d$. and $\nabla ...
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relevant pure math for optimization

I'm interested in optimization generally. I don't know enough to break it down to sub-fields that I would work on. Probability and statistics seem relevant. What about calculus of ...
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A (simple?) question about continuous functions

[EDIT to restrict the function domain to a compact interval, and adapt some of my thoughts.] Suppose I have a continuous and twice differentiable function $f_a(x)$ that maps some compact interval ...
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169 views

Solving a linear programming problem: Are my formulations correct?

QUESTION J (PTY) LTD is a fertilizer manufacturing enterprise that produces two types of fertilizers, namely white and gray. The white fertilizer is for crops like maize, sorghum, etc while the gray ...
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What is the name of this method used to solve a nonlinear problem?

The lecturer taught this method in my Optimization and Control Theory Class and I wasn't quite there when he named it. Could you help me out? He gave the following example of the method in class: ...
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Minimization problem convex set

I'm trying to minimize the function: $$f(w)=w^T\mu+k\sqrt{w^T\Sigma w}$$ where $w$ is a vector in $W=\{x \in \mathbb{R}^n|x_1+...+x_n=1 , x_i \geq 0 \forall i\}$. The vector $\mu \in \mathbb{R}^n$, ...
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Checking whether a solution to MIP is optimal

Consider a binary integer program \begin{align} \min \quad &\sum _{j \in J}f_j x_j +\sum _{i \in I} c_i y_i \notag \\ \mbox{s.t.} \quad &\sum _{j \in N_i} x_j \ge 1-y_i, \quad \forall i\in I ...
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How is it possible to use normals in the definition of a linear programming constraint?

I'm trying to calculate the center of a feasible region in a system of linear inequalities using linear programming techniques. After a bit of research, it looked like defining the center as a ...
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98 views

Local Extrema and Global Extrema

When we have a convex function we know that a local minimum is a global minimum, and similarly for a concave function. What are some other situations where finding local extrema can yield global ...
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Is monotonicity of $g$ necessary and sufficient for preserving critical points of $f$ in $g(f(x))$?

Problem 1. I want to find the argminof $f(x)$. Suppose solving $f'(x)=0$ is too difficult. Instead, solve Problem 2: optimize $g(f(x))$, (presumably by solving $g'(f(x))\cdot f'(x)=0$?). For what ...
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online learning to maximize profit

I have a software which takes input as investment and gives the output as return on a particular stock. Now profit metric $x_i$ is defined as the ratio of return $g_i$ to maximum possible return ...
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70 views

The local minimum of the SQP (sequential quadratic programming) algorithm

Consider the constrained optimization problem \begin{eqnarray} goal~~&&\min f(x)\\ s.t.~~&&g_1(x)\leq0\\ &&g_2(x)\leq0\\ &&\cdots\\ &&g_n(x)\leq0 ...
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186 views

Linear programming simplex - can I have a constraint with a multiplication?

I'm not sure of this, can I have a constraint like this in a linear programming problem to be solved with simplex algorithm? $$n_1t_1 + n_2t_2 > 200$$ where $n_1$ and $t_1$, $n_2$ and $t_2$ are ...
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Operations research - summation notation [duplicate]

Outline: Hermione has been thinking about the imminent return of the Dark Lord, so she has been busy packing her bag with all the items required for her survival. Because she has so many different ...
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62 views

Correlated Equilibrium

I have a question about the definition of the correlated equilibrium. I see that some authors define it as "expected payoff of playing the recommended strategy is no less than playing another ...
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How to ensure maximum reaches certain value

Let us say we have a constant $C$ and variable $t$. There is a function, $p(t)$ whose second derivative is $C-t$. There is also a target maximum called $M$. For what value of $C$, would the maximum of ...
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Numerical Optimization methods?

What kind of functions are suitable for numerical optimization methods such as Newton, Gradient Descent, ... ? Any conditions?
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623 views

Optimization of a function of two variables.

Suppose we have to minimize a function $f(\mathbf x,\mathbf y)$ where $\mathbf x$, $\mathbf y$ are vectors in Euclidean space. The function is convex in $\mathbf y$ when $\mathbf x$ is kept constant ...
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38 views

Minimum point of Constraint set

Let $ c \in R^n $ be non-zero, and consider the problem of minimizing the function $f(x)=c^Tx $ on some constraint set $ S$. Show that a minimum point of this problem cannot lie in the interior of ...
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How to solve this minimization (maximization)?

I'm facing this problem: $$ \large \min_{x \in \mathbb{R}_+^3} \max \left\{ { \sum_{i=1}^3 x_i^2-2 x_1 x_3 \over \left(\sum_{i=1}^3 x_i \right)^2} , { \sum_{i=1}^3 x_i^2 + 2 (x_1 x_3 - ...
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109 views

Given the value at two points of a curve and a constraint on the integral of the curve over the range between them, how do I find the curve itself?

I know two points, say A and B, that are the endpoints of a curve (a regular function) over a certain range. I also have a constraint on the integral over this range, say P. Given only this ...
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Minimizing the time to produce $T$ items with machines that run less efficiently over time

I would like to produce $R$ items in the shortest amount of time possible. For the sake of a visual, call these items bottles of carrot juice. Let $t_1$ be the time to purchase and set up a ...
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488 views

I Need Help Finding the Area of the Largest Trapezoid that can be inscribed in a circle

Im currently learning how to maximize areas. theres a question that I'm stuck on Find the largest trapezoid that can be inscribed in a circle of radius 2 and whose base is the diameter of the ...
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145 views

Mathematical Economics - Utility maximization

I am thankful to any hints: What I have: Simple log-utility form: $u = \log c_1 + \beta \log c_2$ Budget constraints: $c_1 + s \leq w$ $c_2 \leq R\; s$ Problem: For utility maximization: $s = ...
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99 views

Condition or Proof: Minimizer of one function is maximizing another function

I have two real functions $f(X),g(X)$ where the argument $X$ is a real matrix. The solution $X^*$ for the problem of minimizing $f$ is ending up maximizing $g$ as well. I am looking for a way to prove ...
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324 views

How to maximize scalar product using Lagrangian methods

maximize $U(x) = u \cdot x$ with respect to $p \cdot x = w$ given that $u, p, x \in \mathbb{R}^L_+$, $w \in \mathbb{R_+}$. I can solve it classically: \begin{align} u \cdot x &= \sum u_i x_i \\ ...
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A hard multivariate optimization problem in $n-1$ variables

For $n>1$, I want to find the smallest value, and corresponding $x_i$ values, of $f(x_2,\dots,x_n) = \prod_{k=2}^n (x_k+1)^k$ subject to the constraints $x_j > 0$ for all $j$ and $\prod_{k=2}^n ...
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Optimization of entropy for fixed distance to uniform

Suppose that I know that a probability distribution with $n$ outcomes is very close to being uniform (that is: $\forall i,p_i=\frac{1}{n}$), and in particular for $n\epsilon\ll 1$ the distribution ...
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maximization of a Strictly convex function

The things we know, usually minimization of a convex function, unique solution will exist. My question is, maximization of a strictly convex function, will that give an unique maximum? If so how we ...
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88 views

Maximization of minimum difference

Suppose we have a function of the form: $(x_1 - x_2) + (x_3 - x_4) + (x_5 - x_6)$ and we have maximized this summation using linprog (using some constraints which are not important for this matter). ...
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Fastest Algorithm for NLP with linear constraints

I have an minimization problem of the following form: (Im not a mathematician, i come from the programming side, so excuse me if i have not the perfect standard of writing the formulas) $Z(x) = ...
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Solving exponential equations for maxent with opensource applications or coding

I am working on maximum entropy. And I have a problem solving some equations. Look at these 2 equations: $\frac{(2*e^{-a*2-b*2^2}) +(6*e^{-a*6-b*6^2}) +(10*e^{-a*10-b*10^2}) ...
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Prove or disprove that Y = AX-C

Let $A$ be an $m \times n$ matrix such that $\mathrm{rank}(A) = n \le m$. Prove, or disprove using a counter example: Every $m\times n$ matrix $Y$ has a decomposition $Y = AX-C$, where $X$ and ...
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How to maximize $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?

Short Version of the Question: How do I maximize the value of $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$? Long Version of the Question: I'm currently attempting ...
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How to re-parametrize for quadratic minimization?

Given a real-rectangular matrix $S$ and inorder to solve this simple quadratic programming problem: Minimize $w'S'Sw = \|S w\|^2$ over $w$ subject to $e^Tw = 1$ and $w \geq 0$ using a solver I ...
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126 views

How to solve an underdetermined linear system with variables limited to an interval

If I have an underdetermined linear system of equations, with the additional constraint that all of the variables are limited to the interval $[0, 1]$, what techniques are there to solve this in the ...
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Mathematical generalization of the equilibrium point

Let $U$ be an open set $U \subset \mathbb R^n$. Let $f$ be a class-2 function $f: U → \mathbb R$. Prove or disprove the following statement. $∇^2 f=0$ and $∇f= 0$ at $x_0 \in U$ implies $x_0$ is ...
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90 views

Efficient (approximate) projection onto the special orthogonal group

I need to carry out an optimization on the special orthogonal group $SO(n)$. For the line search I use a simple back-projection method $$\mbox{minimize}_\tau f(\pi(X+\tau Z))$$ where $X\in SO(n)$ ...
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Infeasible start Newton's method

I am implementing infeasible start Newton's method from the information in the slides (slide 11 of the link) posted here. It requires us to calculate primal and dual Newton steps, denoted by, $\Delta ...
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on the sufficiency of Lagrange multipliers

Suppose that we have a nonnegative polynomial $f(x) \in \mathbb{R}[x_1,\cdots,x_n]$ and we want to minimize it subject to the polynomial constraint $h(x)=0$ with $h(x) \in \mathbb{R}[x_1,\cdots,x_n]$. ...
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Transpose of matrix inverse: $(AA^T)^{-1}A^Tb \stackrel{?}{=} (A^TA)^{-1}A^Tb$

Given the matrix equation: $$ x^TA^TA = b^TA $$ I'm trying to find the least squares solution (i.e.; trying to minimize $r=||Ax-b||$). The matrix $A$ is not necessarily symmetric. When I solve it ...
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Understanding the Hamiltonian function

Based on this function: $$\text{max} \int_0^2(-2tx-u^2) \, dt$$ We know that $$(1) \;-1 \leq u \leq 1, \; \; \; (2) \; \dot{x}=2u, \; \; \; (3) \; x(0)=1, \; \; \; \text{x(2) is free}$$ I can ...
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Epigraph of a function f: D $\rightarrow$ R is convex iff epif(f) is a subset of D*R which is a convex set

As in the topic, how to show that $epi(f)$ is convex iff $epi(f)$ belongs to D*R which is a convex set.
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52 views

Strict local minimiser

Let $\Omega$ be a convex subset of $R^n$ adm f is a real valued, twice differentiable function. Let $x^*$ to be a point in $\Omega$ and suppose that there exists $c \in R \, c >0$ s.t. for all ...
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Optimization and Rent

The manager of a large apartment complex knows from experience that 110 units will be occupied if the rent is 342 dollars per month. A market survey suggests that, on the average, one additional unit ...
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255 views

Optimization. I need help finding the maximum profit

Josh wants to start a cell phone repair business. Josh determines that $x$ phones can be repaired daily at $p$ dollars per repair, where $x=175-p$. The cost of repairing $x$ phones per day is ...
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68 views

Optimizing the area of a triangle in space.

A triangle has two corners, $(8,0,3)$ and $(0,8,3)$ and a third curve in space that consists of all points $(8,8,a^{2}+3)$, where $a$ is a real number. Calculate the area of the triangle as a function ...
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192 views

3-D Absolute Max/Min over closed&bounded region

Find the absolute max and min values of $f(x,y)=2x+y^2-2$ on the closed and bounded region that lies outside the upper half-circle of $\{(x,y)| x^2+y^2=1\}$, and inside the rectangle given by ...