Questions tagged [bounds]

For questions about obtaining upper or lower bounds for certain values, usually for an optimization objective.

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2
votes
1answer
156 views

How to convexify log(convex) function?

I have the following optimization problem: \begin{align}\max_x&\quad\log_2(1+|a+bx|^2+cx^2)\\\text{s.t.}&\quad0\le x\le1\\&\quad(1-x^2)\ge\text{constant}\end{align} where $a$ and $b$ are ...
1
vote
1answer
48 views

Find an upper bound for an objective function

My objective function is $\log_2(1+{x^2y^2})$ and I found two upper bounds for $x^2$ and $y^2$. For example, assumed that we have the following upper bounds: $x^2\leq\text{constant}_1^2$ and $y^2\leq\...
2
votes
1answer
77 views

Which is better to minimize w.r.t a lower bound or an upper bound of an objective function?

Suppose there is a optimization problem that aims at minimizing an objective function $X$ but we can't develop a mathematical model for minimizing $X$. However, there are two objective functions $Y$ ...
7
votes
3answers
959 views

How do Quadratic Programming solvers handle variables without bounds?

Solvers for non-convex QPs generally do the McCormick relaxation of the term $xy=z$ and then do spatial branch and bound. This requires that $x$ and $y$ have given bounds, how do they handle the case ...
2
votes
0answers
47 views

Heuristic to compute upper bound for the simple assembly line balancing problem type 2?

I am currently working on SALBP-2 described as: Given the number m of assembly stations, minimize the cicle time c. I have done a bit of research and found the following approach to solve it: The ...
2
votes
2answers
167 views

How to use tight upper and lower bounds to get to the optimal value via branch and bound?

I have algorithms that get me a tight upper (UB) and lower (LB) bound to a maximization binary integer program (a routing problem). My formulation is non-compact and requires the addition of sub-tour ...
1
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0answers
70 views

Ways to improve lower bounds while solving MIPs

What are the ways to improve lower bounds while solving a minimization problem (MILP)?
10
votes
2answers
150 views

How to exploit known solution in MILP

I have an MILP model to which I get an integer feasible solution as a result of a heuristic search. In this particular example, the initial solution turns out to be the optimal solution, which I prove ...
7
votes
1answer
70 views

An upper-bound on the value of $S$ in $(s,S)$ policy

I recently have come across a problem which can be categorized as a stochastic optimization. The problem seems simple, but I haven't been able to solve it yet. It has a major impact on algorithm ...
5
votes
1answer
240 views

Solving a variant of multiple knapsack problem/ generalized assignment problem

Consider $m$ knapsack and $n$ items. With each knapsack $j$ associated a capacity $c(j)$ and with each item $i$ associated a profit $p(i,j)$ (that depends on the knapsack, so it's not exactly the ...
17
votes
3answers
816 views

Variable fixing based on a good feasible solution

Suppose you have a combinatorial optimization problem that is formulated as a mixed integer linear program (minimization). The problem size is denoted $n$ and the expected $n$ is around $100$. The ...
11
votes
4answers
339 views

Proof of bound on optimal TSP tour length in rectangular region

Lemma 3 in Haimovich and Rinnooy Kan (1985) (Math of OR 10(4):527–542) says: If $X$ [the set of nodes] is contained in a rectangle with sides $a$ and $b$, then $$ T^*(X) \le \sqrt{2(n-1)ab} + 2(a+...
25
votes
6answers
2k views

How to compare two different formulations of a problem?

I somewhat know how to compare two MILP formulations of a problem that both use the same set of decision variables (as in the classical MTZ vs DFJ formulations of the TSP). I was wondering how two ...
16
votes
1answer
907 views

Prove that these linear programming problems are bounded by $O(k^{1/2})$

Prove that these linear programming problems are bounded by $O(k^{1/2})$ Conjecturally the expanded partial sums of the Möbius transform of the Harmonic numbers have two out of three properties in ...
21
votes
5answers
314 views

Tightness of an LP relaxation without using objective function

How can we measure the tightness of a linear programming relaxation for a mixed integer linear program without using the objective value? I would like to get a measure in terms of the feasible set and ...
15
votes
1answer
507 views

How to get the best bound of large LP problems in CPLEX?

When using the C callable library to solve a large LP, how can I get the best bound after calling the method CPXXlpopt? Does it depend on the algorithm used to ...
27
votes
3answers
529 views

Feeding known lower bounds to solvers

Given an optimization problem that aims at minimizing some objective function, a lower bound that is valid for all optimal solutions, and your solver of choice: For what theoretical and/or practical (...