Equation for optimization problem in linear programming

Suppose that you are trying to solve the optimization problem:

Maximize v⋅x subject to Ax ≥ b for some A∈R^(m×n) (i.e. trying to solve an optimization problem in n variables with m linear inequality constraints).

This problem can be reduced to running a solution finding algorithm on a different system of linear equations in k variables. What is the smallest value of k for which this can be done?

From an optimization perspective, why use the Prestige Ranger?

Unearthed Arcana, and therefore the SRD, contains rules for Prestigious Character Classes. Namely, the Prestige Bard, the Prestige Paladin, and the Prestige Ranger. I have seen the first two crop up in handbooks, answers, and optimization guides from time to time, and not without good reason. However, I cannot recall a single time that I have ever seen the Prestige Ranger recommended, or frankly even mentioned, as being useful for any purpose. Why is this? Does it have some value that I do not know of? If not, what does this class lack that makes its fellow Prestigious Character Classes useful?

Optimization Problem to Maximize Points Given Cost Constraints

I have 2 groups of items: A and B. Each item has an associated cost c i and points p i. I need to choose 3 items from group A, and 2 items from group B such that the sum of points of those 5 items is maximized. There is a constraint where I have limited funds to choose those 5 items. Suppose I have $ 500 to work with, then my constraint is:

$ $ 0 \leq 500 – C_{A1} – C_{A2} – C_{A3} – C_{B1} – C_{B2}$ $

and I want to maximize the sum of their points:

$ $ max = P_{A1} + P_{A2} – P_{A3} – P_{B1} – P_{B2}$ $

For some A1, A2, A3, B1, and B2 provide the maximum # of points. I have the data representing the costs and points for each item.

Thanks in advance.

NP-Complete problem whose corresponding optimization problem is not NP-Hard

For this question I will refer to$ \ NP-hard$ problems as those that are at least as hard as$ \ NP-complete$ problems. That is, a problem$ \ H$ is$ \ NP-hard$ if there is an$ \ NP-complete$ problem$ \ G$ , such that$ \ G$ is reducible to$ \ H$ in polynomial time.$ \ NP-hard$ problems are not restricted to decision problems and are not necessarily in$ \ NP$ .

Considering the above, is there any optimization problem$ \ L$ such that$ \ L \notin NP-hard $ and whose corresponding decision problem is$ \ NP-complete$ ?

For example, consider the case for the travelling salesman problem. (TSP)

Optimization problem: Given a list of cities and the distances between each pair, what is the shortest path that visits each city and returns to the original city?

Decision problem: Given a list of cities, the distances between each pair and a length$ \ L$ , does there exist a path that visits each city and returns to the original city of length at most$ \ L$ .

It is well known that the decision problem of TSP is$ \ NP-complete$ and its corresponding optimization problem is$ \ NP-hard$ .

To sum up, what is an example of a$ \ NP-complete$ problem whose corresponding optimization problem lies outside the class$ \ NP-hard$ ? Perhaps, it is$ \ EXPTIME$ .

SEO optimization errors

Hello everybody..

I get many errors in my website in website who check SEO optimization, Like


what is the different?

I get

  • Description Duplicates
  • H1 Duplicates
  • Title Duplicates
  • Canonical ≠ URL

All these violations on the home page only, how can I solve these problems.

Is dynamic programming restricted to optimization problems?

The usual criteria used to decide if a problem can be solved using dynamic programming is (1) if it has optimal sub-problems and (2) if it has overlapping sub-problems. Does the word “optimal” mean that DP can only be used to solve optimization problems? Otherwise, it would make more sense to write (1) as “if it can be decomposed in sub-problems” (i.e. it is recursive). Obviously, this definition includes optimization problems with optimal sub-problems.

Using decision oracle to solve optimization problem of maximum polyomino tiling

So, this problem is a kind of variant of polyomino packing which has been discussed frequently elsewhere, but I haven’t been able to find anything on my particular problem. Suppose we have a list of polyominos $ p_1, p_2, …, p_n$ (not necessarily distinct), and we want to find a tiling of a rectangle of dimension $ a \times b$ with $ a, b \leq n$ that maximizes the number of squares covered, where we can use each $ p_i$ at most once, and polyominos must be fully contained within the rectangle. Now, we have the decision problem which tells us, for a given $ t$ , if there is some tiling covering at least $ t$ squares, and the optimization problem which is finding a tiling that covers the maximum number of squares. There are two parts: first, if you can solve the optimization problem in polynomial time, can you solve the decision problem in polynomial time? And secondly, if you can solve the decision problem in polynomial problem, can you solve the optimization problem in polynomial time?

If we have an oracle that solves the optimization in polynomial time, solving the decision problem in polynomial time is easy. However, given an oracle for the decision problem, I was unable to find a way to solve the optimization problem in polynomial time. The main issue I’m facing is that the decision oracle only works for rectangular boards, which means we can’t just place pieces and then use the oracle to see if the placement works, since we won’t have a rectangular board if we want to exclude the piece we just placed. It isn’t hard to determine the actual maximum number of tiles you can cover, and you can even find the actual pieces you need to use, but I haven’t been able to figure out a way to find an arrangement of the pieces in polynomial time using the oracle. I assume there is some trick here, but I don’t see it.

Coding optimization

I am wondering if there is a book/course that, instead of giving good written code, it gives a bad/underoptimized code, explain why it’s bad/underoptimized and then gives a better approach.

PS : It can be pseudocode, python or C++ code