# Tagged Questions

Discrete optimization is a branch of optimization in applied mathematics and computer science. As opposed to continuous optimization, some or all of the variables used in a discrete mathematical program are restricted to be discrete variables—that is, to assume only a discrete set of values, such as ...

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### related to biconcave optimization

I have a bivariate function $f(x,y)$ both $x,y$ can assume values within closed interval i.e. $x_1\leq x\leq x_2$ and similarly $y_1 \leq y \leq y_2$. I know that for a fix value of $x$ the function ...
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### Under which conditions discretization of convex\concave function is submodular?

Say, I have $f(x)$ with $x \in [0,1]$, then by discretization I mean $f(x_h)$ with $x_h \in \{0, h, 2h, \dots, 1\}$. I know about Lovasz extension, but it works in other way: given submodular ...
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### Maximum Coin Changes That Does Not Add To a Dollar

What is the maximal amount of money attained from coins of 1, 5, 10, 25 cent denominations that none of its subset amounts to 100 cents? We can find the solution with exhaustive or naive dynamic ...
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### Book Recommendation for Infinite Dimensional Stochastic Optimization Problem in Discrete Time

Let $X(k)$ be i.i.d. discrete random variables and for all $k=0,1,2,...,N-1$, let $X:=(X(0),X(1)...,X(k))$ and $f := (f(0), f(1),...,f(k))$ with $f(k)$ be the decision function at time $k$, I want to ...
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### Optimization of function on integer hypertetrahedron

I have the following optimization problem: Let $k,n \in \mathbb{N}$ with $k < n$. Let $N:=\{1, \dots,n\}$ and $D := \{^{t}(x_1, \dots, x_k) \in N^k \vert \sum_{i=1}^k x_i = N\}$ (hypertetrahedron)...
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### Discrete Approximation to Dynamic Lagrangians

Suppose I have the following dynamic optimization problem, where I want to maximize the function $u(c,h)$ over time that's differentiable in both $c$ and $h$. I'm going to assume that the function $u$ ...
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### Tree-width of a quadratic pseudo-Boolean function

A pseudo-Boolean function $f : \mathbb{B}^n \mapsto \mathbb{R}$ is of the following form. $$f \left(x_1, \ldots, x_n\right) = \sum_{S\subseteq V} c_S \prod_{j \in S} x_j$$ Here $c_S \in \mathbb{R}$,...
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### How to maximize this set function!?

Given a set $F$ and a function $p: 2^F \times 2^F \to [-1,0]$ such that $p (A \cup B, C) \leq p (A,C)$ for any sets $A, B, C \in 2^F$ : Q1: How can we choose a non-empty set $O \in 2^F$ such ...
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### Belt Balancer problem (Factorio)

So this question is inspired by the following thread: https://forums.factorio.com/viewtopic.php?f=5&t=25008 In it, the poster is examining an $8$-belt balancer (more on that to come) which he ...
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### Minimizing the intersection of three sets

Let the sets $A,B,C$ which are all subsets of a larger set $N$. If $N(A), N(B), N(C), N$ are the populations respectively, then i need to find the minimum value of the population of their intersection ...
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### Maximize $\ S=Max{(\sum_{i=1}^{n-1} |f(i+1)-f(i)|)}$

Let $\ f: \{1,2,...n\} → \{1,2,...n\} \quad bijection$ What I want to know is $\ S=Max{(\sum_{i=1}^{n-1} |f(i+1)-f(i)|)}$ Furthermore, Is there a way to know 'when' does S would be maximized? I ...
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### Prove that a function is single peaked

Consider the following function: $$F(K)=\sum_{i=K+1}^{M} P(i,M) - \delta K P(K,M),$$ where $K,M\in \mathbb{N}$, $K<M/2$, $0<\delta,p<1$ and $P(i,M) = \binom{M}{i}p^i(1-p)^{M-i}$. I want to ...