## Contributed Session Wed.2.H 1012

#### Wednesday, 13:15 - 14:45 h, Room: H 1012

**Cluster 17: Nonsmooth optimization** [...]

### Applications of nonsmooth optimization

**Chair: Amirhossein Sadoghi**

**Wednesday, 13:15 - 13:40 h, Room: H 1012, Talk 1**

**Ann-Brith Strömberg**

Lagrangian optimization for inconsistent linear programs

**Coauthors: Emil Gustafsson, Torbjörn Larsson, Magnus Önnheim, Michael Patriksson**

**Abstract:**

When a Lagrangian dual method is used to solve an infeasible optimization problem, the inconsistency is manifested through the divergence of the dual iterates. Will the primal sequence of subproblem solutions still yield relevant information about the primal solution? We answer this question in the affirmative for a linear program and an associated Lagrangian dual algorithm. We show that the primal-dual linear program can be associated with a saddle point problem in which - in the inconsistent case - the primal part amounts to finding a solution in the primal space such that the total amount of infeasibility in the relaxed constraints is minimized; the dual part aims to identify a steepest feasible ascent direction. We present convergence results for a subgradient optimization algorithm applied to the Lagrangian dual problem, and the construction of an ergodic sequence of primal subproblem solutions; this algorithm yields convergence to a saddle point. We establish that the primal sequence finitely converges to a minimizer of the original objective over the primal solutions of the saddle-point problem, while the dual iterates diverge in the direction of steepest ascent.

**Wednesday, 13:45 - 14:10 h, Room: H 1012, Talk 2**

**Adilson Elias Xavier**

The continuous multiple allocation *p*-hub median problem solving by the hyperbolic smoothing approach: Computational performance

**Coauthors: Claudio Gesteira, Vinicius Layter Xavier**

**Abstract:**

Hub-and-spoke (HS) networks constitute an important approach for designing transportation and telecommunications systems. The continuous multiple allocation *p*-hub median problem consists in finding the least expensive HS network, locating a given number of *p* hubs in planar space and assigning traffic to them, given the demands between each origin-destination pair and the respective transportation costs, where each demand center can receive and send flow through more than one hub. The specification of the problem corresponds to a strongly non-differentiable min-sum-min formulation. The proposed method overcomes this difficulty with the hyperbolic smoothing strategy, which has been proven able to solve large instances of clustering problems quite efficiently. The solution is ultimately obtained by solving a sequence of differentiable unconstrained low dimension optimization subproblems. The consistency of the method is shown through a set of computational experiments with large hub-and-spoke problems in continuous space with up to 1000 cities. The quality of the method is shown by comparing the produced solutions with that published in the literature.

**Wednesday, 14:15 - 14:40 h, Room: H 1012, Talk 3**

**Amirhossein Sadoghi**

Piecewise monotonic regression algorithm for problems comprising seasonal and monotonic trends

**Coauthors: Oleg Burdakov, Anders Grimvall**

**Abstract:**

We consider piecewise monotonic models for problems comprising seasonal cycles and monotonic

trends. In contrast to the conventional piecewise monotonic regression algorithms, our algorithm

can efficiently exploit a priory information about temporal patterns.

Our approach is based on establishing monotonic relations between the observations that compose

the data set. These relations make the data set partially ordered, and allows us to reduce the

original data fitting problem to a monotonic regression problem under the established partial order.

The latter is a large-scale convex quadratic programming problem. It is efficiently solved by the

recently developed Generalized Pool-Adjacent-Violators (GPAV) algorithm.