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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Maximize linear equation with 2 variables

How do I maximize the following equation: $$ 150 \le 9.05x + 18.89y \\ \text{constraints: } \\ x > 0, y > 0 \\ \text{$x$ and $y$ must be whole numbers.} $$ I cannot use calculus to solve ...
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Optimization, solving for the 'error' coefficient

Given a modified regression equation: $\hat Y = \exp(\beta_0 + \sum\beta_ix_i + \varepsilon)*F$ where: $\hat Y = 11353$ $\beta_0 = 8.693021$ $\sum\beta_ix_i = 5.95487177696$ $F = 0.21829$ what ...
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Global optimization problem, whats a good approach? Branch and bound method?

Problem: Given a 4-dimensional parameter space. We place bounds on the parameter, say, 0 to 100, so that we have a 4-dimensional rectangle as a search space. We have a cost function that takes the ...
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68 views

difference between linear separable hyperplane and affine separable hyperplane

What is the differece between linear separable hyperplane and affine separable hyperplane? How does one represent a hyperplane graphically in R^2?
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38 views

Confusion regarding ML estimate

I was going through this article and they have this log likelihood given by $$ LL = \sum_{i=1}^n A_i\log p_i + \sum_{i=1}^n A'_i\log(1-p_i).$$ Basically this is the loglikelihood of a logistic ...
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41 views

Discontinuous optimizer but continuous optimal

Consider a locally-bounded, continuous, positive-semidefinite function $f: X \times Y \rightarrow \mathbb{R}_{\geq 0}$, where $X \subset \mathbb{R}^n$ is compact, $Y \subseteq \mathbb{R}^m$. For each ...
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34 views

Optimization of Unconstrained Quadratic form

So I'm learning about optimization of quadratic forms and this textbook goes through definiteness of matrices and principle minors etc. and then goes straight onto optimizing with constraints but ...
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373 views

Linear programming: the optimum of the shortest path problem is attained by $x \in [0, 1]^m$

Let $G=(V,E)$ be a graph, where $|E|=m$, and suppose we formulate the shortest path problem on $G$ as follows: minimize ${}^t(1,\dots,1)x$ such that $Bx={}^t(1,-1,0,\dots,0), x\in \{0,1\}^m$, where $B ...
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value of this possibly Monty Hall-related 2-person zero-sum game?

Player A tosses a fair coin. He knows how it lands; Player B does not. A can now play move 1 or move 2. If he plays move 1, he pays B £1. If he plays move 2, then B can either play move X or move Y. ...
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Bound and minimize a function in two variables

I'm having trouble with the following problem. I am looking at the expression $$ \exp\left(\frac{1}{\lambda}\int_0^t \left(c-\tfrac{1}{2}g(s)\right)^2 \mathrm{d} s\right)\quad (*), $$ where $t>0$ ...
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Does such a bijective mapping exist?

It is a problem I encountered when working on shape optimization(mech eng, not math one). Consider two connected sets $A$ and $B$ in $\mathbb{R}^d$ (it would be nice if $d$ can be chosen arbitrarily, ...
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149 views

Finding competitve equilibrium(consumption rivalry)

Consider two agents (Pascal and Friedman) in a pure exchange economy with two goods and no free disposal. Pascal has a preference relation give by the utility function $$u^P(x_1^P,x_2^P)=a\ln ...
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51 views

consumer utility [on hold]

Suppose a utility function is given by $U(x,y)= \max \{x,y\} + \min \{x,y\}$ $x$ and $y$ are $2$ commodities with prices $1$ and $2$ respectively and consumer's budget is Rs. $150$. What will be the ...
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107 views

Market optimization problem

Demand schedule: $Q_d=a_0-a_1P_d$Supply schedule: $Q_s=b_0+b_1P_s$$P_d$ and $P_s$ are prices faced by consumers and producers. $a_0,a_1,b_0,b_1$ are all positive constants, where $a_0>b_0$. The ...
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145 views

Minimizing a function

For any $\rho$ we want to minimize this function. The minimum of $\pi$ is obtained at a point where $x_2 = \frac{1}{2}x_{1}$ and where $x_1$ minimizes the function defined by $$\begin{cases} ...
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80 views

Optimization Problem with Modulus function

Any ideas how to solve the following problem: $$Minimize: |F(x,y)|+|G(x,y)|$$ s.t. $x<A, y<B$ where $$F(x,y)=ax^2+by^2+cx+dy+e$$ $$G(x,y)=fx^2+gy^2+hx+iy+j$$ and $A,B$ are known constants. Any ...
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Is it possible to calculate weights of a portfolio with negative values?

Sorry in advance if this question is either too basic or really dumb, but I've been researching this and am a bit confused. I'm trying to help my niece with a question she has and the gist of it is ...
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202 views

Nash Equlibria and Maximin Strategies

Consider the following bimatrix game $(2,6)\ \ \ (4,2) \\ (6,0) \ \ \ (0,4) $ I have been asked to compute all equilibria of this game, as well as the maximin strategies for both players. Now I used ...
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157 views

Linear Programming Duality (Basic optimization)

Suppose that $A$ is an $m\times n$ matrix, $D$ is a $p\times n$ matrix, $b$ is an $m$-vector, and $d$ is a $p$-vector. Prove that there does not exist $n$-vector $x$ satisfying $$Ax \geq b, Dx \leq ...
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Is this function convex?

I have a model - function of two vectors $A$ and $B$. I have data that I want to fit to the model and find the model's parameters. The function needs to be convex to find the parameters using ...
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245 views

Portfolio Optimization Problem Without Correlation Info

I received this interesting problem from a friend today: Assume that you are a portfolio manager with $10 million to allocate to hedge funds. The due diligence team has identified the following ...
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65 views

Optimisation Problem

I'm given a lattice with particles having charges which have known magnitude but unknown signs. The primary aim is to stabilize the lattice (or decrease the force acting on the system) by assigning ...
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146 views

Maximum in a cell intersected with a sphere

I have a rectangular cuboid-shaped 3D "cell" with scalar values at each vertex $(v_1,\ldots,v_8)$. Within this cuboid I do tri-linear interpolation. What I want is the maximum value of that function ...
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Differences behind different methods of fminunc in MATLAB?

Assume I have some .m file with a function (and it's gradient) to be used by fminunc() in MATLAB for some unconstrained optimization problem. To solve the problem ...
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245 views

Underlying assumption in a Primal/Dual table

I just read in one of the questions answered by @MikeSpivey that the following table is provided in Sierksma's Linear and Integer Programming: Theory and Practice, Volume 1, page 144. ...
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Basic Optimization Problem

I sat for an exam a few days ago. I managed to answer every question except for question $1$c in the calculus paper. Provided that I got question $2$d correct (my answer was $m=0.5$), the absence of ...
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How do I set up the following problem to arrive at the answer?

A warehouse has 10 unlabelled rows of pallets. Each row of pallets contains thousands of cell phones destined for different countries. Each 100 gram cell phone is exactly the same except for those in ...
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how to compute this optimization problem

Given $A,B$ are positive semidefinite matrices, how to compute $\max_{0\leq P\leq I}\|APBPA\|$, where the norm is spectral norm, i.e. the largest singular value.
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Equivalent optimization problems

I am reading about optimization and I am having difficulty in understanding the following: If a matrix A is $n\times n$ Hermitian, then $\max_{x^{*}x=1} x^{*}Ax$ is solution equivalent to ...
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Relation of $\max_P \; x^T \cdot Py$ and $\min_P \; \|x-Py\|_2$

Consider a permutation matrix $P$ and two vectors $x$, $v$ with 2-norm = 1 and all positive entries. Are the optimal solutions $P^\ast$ of $\max_P \; (x^T \cdot Py)$ and $\min_P \; \|x-Py\|_2$ the ...
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Finding minimum of multidimensional function

My calculus knowledge is pretty limited, but unfortunately I need to solve a problem of the following kind: I'm given a 2 dimensional function $f(x,y)$ from $\mathbb{R}^2$ to $\mathbb{R}$ and I want ...
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Optimization - Get value of Lagrangian

We know that $f(x) \to \min$ subject to $g(x) = t$ and $h(x) \leq m$ can be written as $f(x) + \lambda g(x)\to\min$ subject to $h(x) \leq m$. How do we get value of lambda so that the two problems ...
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Positive semidefinite vector $\bar{x}$ as $\bar{x}>0 :=\bar{x} \lambda \bar{x}^{T}>0$?

$A \lambda A^{T} $ (quadratic form?) is used with matrices to check definiteness. What about with vectors? If I see conditions such as $\bar{x} > 0$, how can I know whether it means $\bar{x}_{i} ...
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Show $\nabla \bar{x}^{T} M \bar{x} = \lambda ( \nabla \bar{x}^{T} \bar{x} )$

I am trying to prove the sentence ...
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Finding the “best” way to map set of points to another set

I've got a set of points (currently 4, but I can increase the number for better accuracy), and I want to find the optimal transformation so that they can be mapped to another set of points. For ...
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Lagrangian dual in continuous domain

The continuous max flow problem is posed as follows : sup $\int_\Omega p_s(x)dx$ subject to : $|p(x)| \le C(x); \forall x \in \Omega $ $p_s(x) \le C_s(x); \forall x \in \Omega $ $p_t(x) \le ...
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Optimization problem with two-step discontinuous function

imagine my function as a staircase with two steps. This function is to be fitted to some empirical data and I'm searching for an algorithm which minimizes the Root Mean Squared Error between this ...
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174 views

Relaxation of a linear constraint in a quadratic programming problem

the problem i have is like following: $x'Qx + f'x \rightarrow \min_x$ subject to $Ax \le 0$. $Q \ge 0$, so there's nothing wrong there, usual QP with a linear constraint. Is there a way to ...
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optimal basic feasible solution and optimal solution

When studying Linear Program, I once meet the following theorem However, the proof given by the notes was not clear to me. I would really appreciated that if you can share with me the insight of ...
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Relation between girth and lower bound of the number of vertices

Let $G$ be a graph with girth $g > 3$. Suppose that every vertex in $G$ has degree at least $k > 1$. Can we found a nice lower bound for $|V(G)|$? Let me be more specific on what I want: ...
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Determining initial values for optimization problem

I am trying to solve an optimization problem with a quadratic objective function and non-linear constraints, using SQP (Sequential Quadratic Programming). I am attempting at doing the implementation ...
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Get number equation using specific set of values for get given answer

I have do it for AI assignment. Need a logic for finding solution ..Here is the explanation of problem . I have answer ( any number like for example 10 ). And have some set of numbers (like for ...
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How to interpret a discontinuity in 2D Pareto Frontier?

I've solved a bi-objective optimization problem by means of NOMAD solver from OPTI Toolbox and as a result I've obtained a Pareto frontier: How to interpret the visible "gap" in the Pareto frontier? ...
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Upper bound on optimal multinomial logit

Let $[N]={1,...,N}$ denote a set of items, item $i$ has a unit revenue of $r_i>0$ and a utility $u_i>0$. Items have to be assorted in $N$ slots with sampling probabilities $v_k>0$. Let ...
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How exactly do I prove that I find the maximum of the function

I am currently trying to maximize an objective function $f(a,b,c,d,e)$ over the variable $b$ only. By taking the derviative of f over b, setting it to zero, I can solve b in terms of the other 4 ...
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Minimizing sum of functions implies minimizing their squares, maximizing the sum of the inverses?

I have $n$ functions (Say $f_1\space to \space f_n$) of $k$ variables (Say $x_1\space to\space x_k$) each. The functions are all positive, as well as the variables $xi's$. I do not have explicit ...
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Strategies to work with system of trigonometric inequality

I'm trying solve this problem using matlab, anybody know good strategies to work with system of trigonometric inequalities such as $ ...
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sum of logarithms of linear-fractional functions Optimization Problem

I am new to optimization theory and I am facing this optimization problem. \begin{equation} maximize \qquad f(x) = \sum_{i} ...
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multivarable optimization problem, what is the procedure?

Sorry for this obvious question. I am trying to maximize an objective function that consist of 5 variables (a,b,c,d,e) over a and b. That is , $max _{a,b}f(a,b,c,d,e).$ So I procedure I took is ...
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Volume of a polytope cut off by a hyperplane

Given a maximization problem with constraints, and adding a few more constraints using the Gomory cuts and solving the relaxed maximization problem, we can arrive at integer solutions. I am looking to ...