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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To show $f(x)$ has ONLY one Max in $x\in[0,1]$

I have function $$f(x)=\left(\frac{1-x}{2-x}\right)x^{p-1}~\text{where}~p>1;~x\in[0,1]$$ I want to show that $f(x)$ has ONLY one maximum in $x\in[0,1]$. I get the second derivative as ...
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How find this minimum

Help me! Let $x,y,z\ge0$ such that: $xy+yz+zx=1$. Find the minimum value of: $A=\dfrac{1}{x^2+y^2}+\dfrac{1}{y^2+z^2}+\dfrac{1}{z^2+x^2}+\dfrac{5}{2}(x+1)(y+1)(z+1)$ I found minimum value of $A$ ...
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Max-Min optimization problem with $N$ entries

I want to find the optimal $x$, say $x^*$, which maximizes the minimum of $N$ entries as given below: \begin{equation} \begin{split} &\max_{x}~\min ...
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24 views

Minimum Volume of a circular, right cone, with a sphere inscribed in it.

Question: A sphere of radius $r$ is inscribed in a circular, right cone. What is the minimum radius and height of the circular cone? (Thus, volume) Because the answer would specifically ...
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1answer
71 views

No clear analytic method to prove unique maximum? ($2^{-x}+2^{-1/x}$)

Prove that $f(x) = 2^{-x}+2^{-1/x}$ has the unique local maximum $(1,1)$ for $x>0$. Do not use computer software. Proving that $(1,1)$ is a maximum is easy, but I'm having trouble with the ...
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Augmented Lagrangian Method for Inequality Constraints

Augmented Lagrangian Method can be used with inequality constraints. One approach (according to Nocedal book page 522), is linearly constrained Lagrangian. Description is shown in the attached image. ...
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21 views

How to prove a solution is indeed a constrained minimum?

I'm reading the following example on Heath's Scientific Computing (page 266, second edition if anyone has it). "Minimize $f(x_1,x_2)=2\pi x_1(x_1+x_2)$ subject to $g(x_1,x_2)=\pi x^2_1x_2-V$" ...
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1answer
25 views

Envelope Theorem and Static Optimization

The Statement of the Problem: For fixed $r \gt 0$ and $m$, find the maximum value of $1-rx^2-y^2$ on the constraint set $x+y=m$. Find the value function $f^*(r,m)$ and compute $\frac{\partial ...
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Optimal Matching Distance

I'm stuck on problem II.5.9 from Bhatia's Matrix Analysis. The problem is as follows: Let $\{\lambda_1,\dots,\lambda_n\},\{\mu_1,\dots,\mu_n\}$ by two $n$-tuples of complex numbers. Let $$ ...
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1answer
25 views

Non-uniqueness of worst-case (max-min or min-max) optimization

I have a worst-case optimization problem, where i want to maximize the minimum from the uncertainty set (uncertainty is given as an ensemble of 100 realizations, so an ensemble based approach). It is ...
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Formulate an optmization problem as a convex optmization problem

Let $P$ be a polyhedron, i.e. $P = \{ x \in \mathbb{R}^{n}\, |\,\, a_{i}^{T}x \leq b_{i} \}$. Define $R$ as the rectangle given by $\{ x \in \mathbb{R}^{n}\, \mid\, \, l \preceq x \preceq u \}$. Find ...
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Assignment problem, minization of the Standard Deviation

I have an assignment problem. So typically I need to find the optimal combination between two sets of parameters P, M. I know that the Hungarian Algorithm is often privileged for this kind of problem ...
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1answer
100 views
+50

Maximize the largest eigenvalue of a Hermitian matrix constrained by quadratic polynomials

I am looking for a method to maximize under $\mathbf{y}$ the largest eigenvalue of the following Hermitian matrix \begin{equation} S = \left [ \begin{array}{ccc} \mathbf{y}^{H}S_{11}\mathbf{y} ...
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1answer
12 views

Dimensions of a paddock (3 sided rectangle) to enclose maximum possible area

I need help with Qs 4, 5 and 6!! Three sides of a rectangular paddock are to be fenced, the fourth side being an existing straight water drain. If 1000m of fencing is available, what dimensions ...
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1answer
40 views

what is the optimal matrix solution for this problem?

i want to maximize this objective function: $$j(W)=\frac{\mathrm{tr}(WAW^T)}{\mathrm{tr}(WBW^T)}$$ Where $W$ is a $f\times m$ matrix and the matrices $A, B$ have size $m\times m$ and $\mathrm{tr}$ is ...
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1answer
346 views

Linear Programming with One Quadratic Equality Constraint

I have a problem which can be formulated as a Linear Programming with One Quadratic Equality Constraint: where variable x is n-dimensional vector and H is a Semi-Positive Definite n-by-n matrix. I ...
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1answer
347 views

Derivation of Euler-Lagrange equation

Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation. If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is $\dfrac{\partial ...
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6 views

Spin-off of Scheduling Weighted Interval Problem

I'm trying to solve a problem in which, given a + sign shaped area of land (with no width) and a list of contiguous sections of the land (segments, T-shapes, smaller + shapes, etc), each with an ...
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2answers
28 views

Find Algorithm, given a list of arcs, that maximizes number that fit on a circle

I'm trying to find an optimal algorithm that, given a list of arcs $(x_i, y_i)$, where $x_i$ and $y_i$ are the starting and ending angle measurements of the arc in radians, maximizes the number of ...
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3answers
62 views

Question about maximizers and trig

Hi there I have a quick question about the following Consider the simple maximization problem of $$f(x,y)= \frac{x}{1+x^2+y^2}$$ It can be easily seen from analysis of critical points obtained from ...
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17 views

Algorithm to find value where complex numbers meet on the unit circle. [on hold]

I'm trying to find the value at which 4 points are meeting on the unit circle. These points are eigenvalues of the translation operator $T$. By varying $\lambda$ the eigenvalues change. Background: ...
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Linear programming (or possibly nonlinear) formulation

The problem is like this; The construction company is considering erecting three office buildings. The time required to complete each of them and the number of workers required required to be on the ...
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18 views

bound on Lagrange multipliers

Under what conditions is it possible to bound the Lagrange multipliers in a given optimiztion with constrains problem?
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47 views

Shortest line segment that is cut off by the first quadrant and passes through a given point

Let $a$ and $b$ be two positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point $(a,b)$. I attempted to let the ...
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2answers
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compute the smallest affine subspace containing $S$, where $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ is a set of vectors in $\mathbb R^3$

I've started to study convexity to enchance my optimization skills. Given a set $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ of vectors in $\mathbb R^3$ an exercise asks to compute the smallest affine ...
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1answer
43 views

$\arg\max$ of an increasing function grows as the region grows.

$f_1,\dots,f_N:\mathbb{R}^+\rightarrow\mathbb{R}^+$ are strictly increasing, bounded functions whose derivatives monotonically decrease to $0$ as their argument increases. (Picture the shape of the ...
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1answer
58 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
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1answer
29 views

A problem on finding the nearest points to the origin on the intersection of two surfaces

Suppose we are to find the points nearest to the origin on the curve of intersection of the two surfaces $g^{-1}_{1}\{ 0 \}$ and $g_{2}^{-1}\{ 0 \}$, where $g_{1}: (x, y, z) \mapsto x^{2} - xy + y^{2} ...
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converting a equation to convex form which can be given to cvx solver to solve it. [on hold]

Can anybody tell me how to convert this to quadratic programming format so that CVX could solve it...?? I am not asking the whole solution but need only conversion. objective is:- minimize {sum ( ...
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1answer
70 views

Dimension of garden to minimize cost

Math question: A homeowner wants to build, along her driveway, a garden surrounded by a fence. If the garden is to be $5000$ square ft, and the fence along the driveway cost $6$ dollars per foot while ...
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2answers
59 views

Confounding Lagrange multiplier problem

Optimize $f(x,y,z) = 4x^2 + 3y^2 + 5z^2$ over $g(x,y,z) = xy + 2yz + 3xz = 6$ According to the theorem the gradients must be parallell, $\nabla f = \lambda \nabla g$, so their cross product must ...
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1answer
299 views

Smooth approximation of maximum using softmax?

Look at the Wiki page for Softmax function (section "Smooth approximation of maximum"): https://en.wikipedia.org/wiki/Softmax_function It is saying that the following is a smooth approximation to the ...
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Is optimal solution to dual not unique if optimal solution to the primal is degenerate?

If optimal solution to the primal is degenerate, does it necessarily follow that optimal solution to dual not unique? That is, is uniqueness an unnecessary assumption? Spin-off from here. In my ...
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Epsilon constraint method - Pareto optimal solution representation

There's a course that I do remotely and I have a homework question which I have no idea how to answer. I did look up a lot in google and did not find any good examples - only loads of information and ...
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2answers
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Why is the gradient of the objective function in the Lagrange multiplier theorem not $= 0$?

A special case of the Lagrange multiplier theorem may be stated as: Let $S, T \subset \mathbb{R}^{n}$ be open. Let $f: S \to \mathbb{R}$ be differentiable on $S$ and $g: T \to \mathbb{R}$ ...
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Positive solutions to $A^T A x \geq 0$ [on hold]

Find a positive solution $x$ to the linear inequality $A^T A x \geq 0$. $A$ is an arbitrary matrix. I was wondering if there is a general solution. EDIT: One special solution is when $A^TA$ is row ...
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5answers
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Maximum value of $ x^2 + y^2 $ given $4 x^4 + 9 y^4 = 64$ [closed]

It is given that $4 x^4 + 9 y^4 = 64$. Then what will be the maximum value of $x^2 + y^2$? I have done it using the sides of a right-angled triangle be $2x , 3y $ and hypotenuse as 8 .
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319 views

De Jong's Fifth Function's Minimum?

What is the minimum solution to De Jong's fifth function, in the range $-65.536\leqslant x_1\leqslant 65.536, -65.536\leqslant x_2\leqslant 65.536$?
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Mixture of Maximum Entropy and Minimum Cross Entropy?

Assume you have a discrete prior distribution on a set of points $ P(X=\{0,3,5,6\}) = (.40,.30,.20,.10)$ $E[X]=5/2$ And you want to create a new distribution, $Y$, on $\{0,1,2,3,4,...\}$ using the ...
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1answer
27 views

maxima minima optimization problem

My problem is Find the greatest and least distance of the surface $6x^2+4xy+3y^2+14z^2=14$ from the origin. I know that mathematical model of problem is $f(x,y,z)=x^2+y^2+z^2$ subject to ...
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1answer
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What would be the basic solution of this maximization problem? [on hold]

Maximize $P=40x_1+50x_2$ Subject to $x_1+6x_2 \leq 72$ $x_1+3x_2 \leq45$ $x_1, x_2 \geq0$
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1answer
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primal to dual solution conversion ??

i have an optimization problem $$\text{ maximize } z=3x+4y$$ $$\text{ such that: } x+y ≤ 450 \text{ and } 2x+y ≤ 600$$ the optimal solution to this problems comes to be $x=0$; $y=450$; $p=150$ ...
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minimising multivariate quadratic function over integer variables

I have a quadratic function $x_1^2+x_2^2-(u_1x_1+u_2x_2)^2$ which I need to minimise over integer $x_1$ and $x_2$; also, the coefficients $u_1,u_2<1$. In other word, assuming coefficients ...
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1answer
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An LP problem from David G. Luenberger's Linear and Nonlinear Programming book

Could someone help me to solve the following problem? A class of piecewise linear functions can be represented as $f(x) = Maximum (c_{1}^Tx+ d_{1}, c_{2}^Tx, \cdots, c_{p}^Tx + d_{p})$. For such a ...
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1answer
44 views

A maximization problem parametrized by a function

Let $f$ be a smooth positive monotonically increasing real function which is defined and finite in $[0,1]$, and define the following two quantities (see the figure below): $F=\int_{x=0}^1{f(x)dx}$ = ...
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2answers
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How to minimize $w^{T}Aw$?

$A$ is $n\times n$ matrix. Find a $w$ ( $n$-dimensional unit vector) which minimizes this function. By $w^{T}$, I mean $w$-transpose. I understand there would be non-linear optimization techniques ...
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maxima minimum problem 7 [on hold]

Find the maximim and minimum value of the function $f(x,y)=(x+1)^2+y^2$ on the part of the graph of $y^2-x^3=0$ from $(1,-1)$ to $(1,1)$ Can someone help?
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Minimizing Sum of Least Squares in Matlab

I am working on this minimization problem for image warping that I want to solve in Matlab: Each feature $p$ can be presented by a 2D bilinear interpolation of the four vertices $V_p = [v_p^1, ...
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1answer
19 views

How to optimize these parameter

How to optimize the following respect to lambda1 and lambda2: $\sum_{i} f(i) * log(\lambda_1 g(i) + \lambda_2 h(i))$ f(i), g(i), h(i) are known funtions Find lambda1 and lambda2 that satisfy ...
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2answers
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A differential maximization problem

OK, I know how to solve maximization problems on numbers, and I know how to solve differential equations which are equations on functions, but how do I solve a 'maximization problem' over functions? ...