The Lagrangian dual of an integer program can be formulated as a min-max problem where the objective function is convex, piecewise affine and, hence, nonsmooth. It is usually tackled by means of subgradient algorithms, or multiplier adjustment techniques, or even more sophisticated nonsmooth optimization methods such as bundle-type algorithms. Recently a new approach to solving unconstrained convex finite min-max problems has been proposed, which has the nice property of working almost independently of the exact evaluation of the objective function at every iterate-point. In the paper we adapt the method, which is of the descent type, to the solution of the Lagrangian dual. Since the Lagrangian relaxation need not be solved exactly, the approach appears suitable whenever the Lagrangian dual must be solved many times (e.g., to improve the bound at each node of a branch-and-bound tree), and effective heuristic algorithms at low computational cost are available for solving the Lagrangian relaxation. We present an application to the Generalized Assignment Problem (GAP) and discuss the results of our numerical experimentation on a set of standard test problems.

### On solving the Lagrangian dual of integer programs via an incremental approach

#### Abstract

The Lagrangian dual of an integer program can be formulated as a min-max problem where the objective function is convex, piecewise affine and, hence, nonsmooth. It is usually tackled by means of subgradient algorithms, or multiplier adjustment techniques, or even more sophisticated nonsmooth optimization methods such as bundle-type algorithms. Recently a new approach to solving unconstrained convex finite min-max problems has been proposed, which has the nice property of working almost independently of the exact evaluation of the objective function at every iterate-point. In the paper we adapt the method, which is of the descent type, to the solution of the Lagrangian dual. Since the Lagrangian relaxation need not be solved exactly, the approach appears suitable whenever the Lagrangian dual must be solved many times (e.g., to improve the bound at each node of a branch-and-bound tree), and effective heuristic algorithms at low computational cost are available for solving the Lagrangian relaxation. We present an application to the Generalized Assignment Problem (GAP) and discuss the results of our numerical experimentation on a set of standard test problems.
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2009
Incremental algorithms; Convex minimization; Finite min-max
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.11770/139755`
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