User star - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T11:43:28Z http://mathoverflow.net/feeds/user/21172 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/123730/functional-maximization functional maximization Star 2013-03-06T10:03:12Z 2013-03-06T10:03:12Z <p>Define a functional space of functions of the form $F(t)=p_1 exp^{-\mu_1(\delta-t)}+p'_1 (1-exp^{-\mu_1(\delta-t)}))$. $p_1,p'_1,\delta,\mu$ are parameters in [0,1] and trivially, variation of these parameters builds the instances of our functional space. The goal is to solve the following optimization problem tractably:</p> <p>$max_{F_1,F_2\in F(t)} \int_0^\delta \lambda exp^{-\lambda t} (F_1(t)-F_2(t))dt$.</p> http://mathoverflow.net/questions/102566/solving-a-system-of-linear-inequalities Solving a system of linear inequalities Star 2012-07-18T17:32:27Z 2012-07-18T20:36:26Z <p>Consider the following system of inequalities:</p> <p>$Ax=b$; $x\geq 0$;</p> <p>A is a $m\times n$ (non-square) and sparse matrix in which some part of entries are rational. How this system can be solved without using linear programming? </p> http://mathoverflow.net/questions/102533/linear-programming-boundedness Linear programming boundedness Star 2012-07-18T12:09:37Z 2012-07-18T12:09:37Z <p>Assume we are given a linear programming problem. It is well-known that a linear programming problem is unbounded iff there exists an EXTREME direction $d$ such that cd>0 (consider maximization case). Now, I want to check whether my LP is unbounded. I described the existence of a recession direction in the feasible area of the given linear programming problem using constraints 2,3 and constraint (1) is for unboundedness.</p> <p>1) $cd>0$;</p> <p>2) $Ad \leq 0$;</p> <p>3) $Bd \leq 0$;</p> <p>After checking the feasibility of the above polyhedral, I have two directions. First, the polyhedral is infeasible. In this case for sure the problem is bounded (Since I could not find any feasible and also basic feasible solution for the corresponding LP). Second, what if the polyhedral is feasible? Then, can I say that I have an EXTREME ray $d$ with $cd>0$ ?</p> http://mathoverflow.net/questions/102023/large-scale-sparse-system-of-linear-equations Large scale sparse system of linear equations Star 2012-07-12T07:52:04Z 2012-07-12T07:52:04Z <p>What is the best know algorithm for solving a large sparse system of linear equations? The system I'm working on is not symmetric, not positive definite and integer. The only benefit is being sparse. I also need to point out that the matrix is not square. The dimension is $m\times n$ and it is not generally either underestimate or overestimate.</p> http://mathoverflow.net/questions/100350/consistency-of-a-system-of-linear-equations Consistency of a system of linear equations Star 2012-06-22T12:35:59Z 2012-06-22T12:35:59Z <p>I have a system of linear equations in form of $AX=b$ where $A_{m\times n}$, $X_{n\times 1}$ and $b_{m\times 1}$. Coefficient matrix $A$ is quite sparse. However, using a practical LP solver like LINGO it is clear that after permutation of rows, it turns out to be like a lower triangular matrix. I do not know what the permutation function should be. Since the matrix dimension is not square, I cannot use the LU decomposition to solve the system efficiently. Can you please let me know an efficient method for linear systems with non-square coefficient matrix?</p> http://mathoverflow.net/questions/123730/functional-maximization Comment by Star Star 2013-03-06T15:05:23Z 2013-03-06T15:05:23Z You know the general forms of $F_1$ and $F_2$ functions. How do you integrate over unknown instances of the functional space? http://mathoverflow.net/questions/102566/solving-a-system-of-linear-inequalities Comment by Star Star 2012-07-18T17:51:54Z 2012-07-18T17:51:54Z I meant, how can the feasibility of this system be checked without using linear programming? http://mathoverflow.net/questions/102333/linear-programming-boundedness Comment by Star Star 2012-07-16T11:16:08Z 2012-07-16T11:16:08Z The correctness of this statement may help me to simplify a long proof. http://mathoverflow.net/questions/87697/lp-infeasibility Comment by Star Star 2012-02-06T18:15:44Z 2012-02-06T18:15:44Z [Continuation] Now, I got an approximate optimal solution vector for the converted LP in which values for each pairs of variables are within $\epsilon$ of each other. This solution is not feasible for the original LP but still is optimal based on the variable transformation rules that I have used to show the equivalence of these two LP's. Now, the problem is pushing this solution to be feasible by not running the objective value too much. http://mathoverflow.net/questions/87697/lp-infeasibility Comment by Star Star 2012-02-06T18:10:43Z 2012-02-06T18:10:43Z Consider the following original LP: min c'x s.t: Ax=0 0&lt;=x&lt;=1 This is my original LP which has to be solved. Now, using some reductions, I reduced the original LP to the following converted LP: min d'y s.t: By=0 0&lt;=y&lt;=1 Using some relations between x and y, I proved that these two LP are equivalent. It means the optimal solution for one of them gives the optimal solution for the other and visa verse. For the optimal solutions the objective value are the same as well. Now, the converted LP cannot be solved exactly. [Continuation in the next comment]. http://mathoverflow.net/questions/87650/how-to-modify-a-sequence-of-numbers Comment by Star Star 2012-02-06T17:08:53Z 2012-02-06T17:08:53Z Sorry. Just assume they are within \epsilon of each other.