Abstract. Lagrangian duality in mixed integer optimization is a useful framework for problems decomposition and for producing tight lower bounds to the optimal objective, but in contrast to the convex counterpart, it is generally unable to produce optimal solutions directly. In fact, solutions recovered from the dual may be not only suboptimal, but even infeasible. In this paper we concentrate on large scale mixed-integer programs with a specific structure that is of practical interest, as it appears in a variety of application domains such as power systems or supply chain management. We propose a solution method for these structures, in which the primal problem is modified in a certain way, guaranteeing that the solutions produced by the corresponding dual are feasible for the original unmodified primal problem. The modification is simple to implement and the method is amenable to distributed computations. We also demonstrate that the quality of the solutions recovered using our procedure improves as the problem size increases, making it particularly useful for large scale instances for which commercial solvers are inadequate. We illustrate the efficacy of our method with extensive experimentations on a problem stemming from power systems.

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In this paper, we present a new method for finding robust solutions to mixed-integer linear programs subject to uncertain events. We present a new modeling framework for such events that result in uncertainty sets that depend parametrically on the decision taken. We also develop results that can be used to compute corresponding robust solutions. The usefulness of our proposed approach is illustrated by applying it in the context of a scheduling problem. For instance, we address uncertainty on the start times chosen for the tasks or on which unit they are to be executed. Delays and unit outages are possible causes for such events and can be very common in practice. Through our approach, we can accommodate them without altering the remainder of the schedule. We also allow for the inclusion of recourse on the continuous part of the problem, that is, we allow for the revision of some of the decisions once uncertainty is observed. This allows one to increase the performance of the robust solutions. The proposed scheme is also computationally favorable since the robust optimization problem to be solved remains a mixed-integer linear program, and the number of integer variables is not increased with respect to the nominal formulation. We finally apply the method to a concrete batch scheduling problem and discuss the effects of robustification in this case.Communicated by Christodoulos Floudas.

This paper presents an innovative method to achieve the simultaneous phase control of multiple resonance frequencies of a linear multi-degree-of-freedom system using only one sensor/actuator pair. Each frequency is manipulated independently by means of a PLL-based control loop which comprises a digital averaging phase detector that combines the two most crucial tasks, namely the phase shift measurement and the frequency separation. The properties of the controller are designed individually for each mode using a linearized model of the control system. The method is applicable to all kinds of oscillators where frequencies near the structure’s natural frequencies are to be controlled. To validate the results experimentally, the controller is implemented on a digital signal processor (DSP) and applied to a torsional oscillator. Investigating two different damping conditions, the simultaneous control of five resonance frequencies of the oscillator illustrates the effectiveness and stability of the multiple frequency tracking. The method is able to significantly improve the accuracy and versatility of sensor applications. As an example a method is presented that enables the direct determination of the modal damping by using two frequencies corresponding to one single vibration mode.