# Questions tagged [bounds]

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

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### Understanding the condition of the bounded variable algorithm in the linear programming

Following is the section 7.3 of Operation Research An Introduction by Hamdy A. Taha, Define the upper-bounded LP model as, $$\max z=\{CX|(A,I)X=b,0\leq X\leq U\}$$ The bounded algorithm uses only the ...
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### Find projection onto implicitly defined set

I think this is a problem a lot of people have in minimization but I couldn't find algorithmic approaches to it. Given a closed domain $D\subset R^n$ over which a function $f$ is supposed to be ...
1 vote
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### Get objective upper bound in a callback (cplex)

I wonder if there is a way to retrieve the global objective upper bound in a callback in Cplex (in cpp)? I tried several context/IloCplex/IloModel... methods but nothing worked.
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### Deriving a lower bound for a two-stage stochastic problem

Assume an inventory stochastic optimization problem in the following form: $$\min\limits_{x\in X} c^\top x + \mathbb{E}_{\mathbb{\xi}}[\mathcal{Q}(x, \xi)]$$ Demand is the uncertain parameter, and is ...
333 views

### Upper bound on length of solution of linear program

Consider the linear program: $$\text{maximize} ~~ c\cdot x \\ \text{subject to} ~~ A\cdot x\leq b, ~~x\geq 0.$$ Suppose $A$ is an $m\times n$ matrix, $b$ an $m\times 1$ vector and $c$ an $n\times 1$ ...
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### Reduced cost fixing for binary programs

Consider the binary program $$\min_{ x \in \{0,1\}^N } \left\{ c^T x \mid Ax \leq b \right\}$$ where $A$ and $b$ are real matrices with appropriate dimensions. I am interested in solving large binary ...
258 views

### Binary decision variable to indicate whether a continous decision variable is equal to its upper bound

Given a continuous nonnegative decision variable $x\in [0,T]$ bounded by $T$, how can we enforce a relation between $x$ and another binary decision variable $y$ such that when $x$ is equal to its ...
239 views

### Improving best bound within B&B process

Here is an extract from the BnB solution process of my problem. The solver determines a value of 2627.452494 as being the best bound of the optimal solution. The value for the best bound remains the ...
188 views

### What is the role of computation lower bounds for exact methods, heuristics, and hybrid of exact and heuristics?

I'm struggling with this question for weeks: What is the main difference between the role of computation lower bounds for exact methods, heuristics, and hybrid of exact and heuristics? I try to answer ...
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926 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 $Y$ ...
1k 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 ...
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### 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 ...
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### 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 ...
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### Ways to improve lower bounds while solving MIPs

What are the ways to improve lower bounds while solving a minimization problem (MILP)?
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### 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 ...
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### 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 ...
468 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 ...
1k 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 ...
384 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+...
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### 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 ...
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### 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 ...
632 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 ...