A Class of Mathematical Programs with Equilibrium Constraints: A Smooth Algorithm and Applications to Contact Problems

We discuss a special mathematical programming problem with equilibrium constraints (MPEC), that arises in material and shape optimization problems involving the contact of a rod or a plate with a rigid obstacle. This MPEC can be reduced to a nonlinear programming problem with independent variables and some dependent variables implicity defined by the solution of a mixed linear complementarity problem (MLCP). A projected-gradient algorithm including a complementarity method is proposed to solve this optimization problem. Several numerical examples are reported to illustrate the efficiency of this methodology in practice

A reformulation–linearization–convexification algorithm for optimal correction of an inconsistent system of linear constraints

Abstract In this paper, an algorithm is introduced to find an optimal solution for an optimization problem that arises in total least squares with inequality constraints, and in the correction of infeasible linear systems of inequalities. The stated problem is a nonconvex program with a special structure that allows the use of a reformulation-linearization-convexification technique for its solution. A branch-and-bound method for finding a global optimum for this problem is introduced based on this technique. Some computational experiments are included to highlight the efficacy of the proposed methodology

On the Quadratic Eigenvalue Complementarity Problem

Abstract We introduce several new results on the Quadratic Eigenvalue Complementarity Problem (QEiCP), focusing on the nonsymmetric case, i,e, without making symmetry assumptions on the matrices defining the problem. First we establish a new sufficient condition for existence of solutions of this problem, which is somewhat more manageable than previously existent ones. This condition works through the introduction of auxiliary variables which leads to the reduction of QEiCP to an Eigenvalue Complementarity Problem (EiCP) in higher dimension. Hence, this reduction suggests a new strategy for solving QEiCP, which is also analyzed in the paper. We also present an upper bound for the number of solutions of QEiCP and exhibit some examples of instances of QEiCP whose solution set has large cardinality, without attaining though the just mentioned upper bound. We also investigate the numerical solution of the QEiCP by exploiting a nonlinear programming and a variational inequality formulations of QEiCP. Some numerical experiments are reported and illustrate the benefits and drawbacks of using these formulations for solving the QEiCP in practice

Relative sit-to-stand power: aging trajectories, functionally relevant cut-off points, and normative data in a large European cohort

Background: A validated, standardized, and feasible test to assess muscle power in older adults has recently been reported: the sit-to-stand (STS) muscle power test. This investigation aimed to assess the relationship between relative STS power and age and to provide normative data, cut-off points, and minimal clinically important differences (MCID) for STS power measures in older women and men. Methods: A total of 9320 older adults (6161 women and 3159 men) aged 60–103 years and 586 young and middle-aged adults (318 women and 268 men) aged 20–60 years were included in this cross-sectional study. Relative (normalized to body mass), allometric (normalized to height squared), and specific (normalized to legs muscle mass) muscle power values were assessed by the 30 s STS power test. Body composition was evaluated by dual energy X-ray absorptiometry and bioelectrical impedance analysis, and legs skeletal muscle index (SMI; normalized to height squared) was calculated. Habitual and maximal gait speed, timed up-and-go test, and 6 min walking distance were collected as physical performance measures, and participants were classified into two groups: well-functioning and mobility-limited older adults. Results: Relative STS power was found to decrease between 30–50 years (-0.05 W·kg-1·year-1; P > 0.05), 50–80 years (-0.10 to -0.13 W·kg-1·year-1; P < 0.001), and above 80 years (-0.07 to -0.08 W·kg-1·year-1; P < 0.001). A total of 1129 older women (18%) and 510 older men (16%) presented mobility limitations. Mobility-limited older adults were older and exhibited lower relative, allometric, and specific power; higher body mass index (BMI) and legs SMI (both only in women); and lower legs SMI (only in men) than their well-functioning counterparts (all P < 0.05). Normative data and cut-off points for relative, allometric, and specific STS power and for BMI and legs SMI were reported. Low relative STS power occurred below 2.1 W·kg-1 in women (area under the curve, AUC, [95% confidence interval, CI] = 0.85 [0.84–0.87]) and below 2.6 W·kg-1 in men (AUC [95% CI] = 0.89 [0.87–0.91]). The age-adjusted odds ratios [95% CI] for mobility limitations in older women and men with low relative STS power were 10.6 [9.0–12.6] and 14.1 [10.9–18.2], respectively. MCID values for relative STS power were 0.33 W·kg-1 in women and 0.42 W·kg-1 in men. Conclusions: Relative STS power decreased significantly after the age of 50 years and was negatively and strongly associated with mobility limitations. Our study provides normative data, functionally relevant cut-off points, and MCID values for STS power for their use in daily clinical practice. © 2021 The Authors. Journal of Cachexia, Sarcopenia and Muscle published by John Wiley & Sons Ltd on behalf of Society on Sarcopenia, Cachexia and Wasting Disorders

A Complementarity-based Partitioning and Disjunctive Cut Algorithm for Mathematical Programming Problems with Equilibrium Constraints

Abstract In this paper a branch-and-bound algorithm is proposed for finding a global minimum to a Mathematical Programming Problem with Complementarity (or Equilibrium) Constraints (MPECs), which incorporates disjunctive cuts for computing lower bounds and employs a Complementarity Active-Set Algorithm for computing upper bounds. Computational results for solving MPECs associated with Bilivel Problems, NP-hard Linear Complementarity Problems, and Hinge Fitting Problems are presented to highlight the efficacy of the procedure in determining a global minimum for different classes of MPECs

A block active set algorithm for large-scalequadratic programming with box constraints

Abstract An algorithm for computing a stationary point of a quadratic program with box constraints(BQP) is proposed. Each iteration of this procedure comprises a guessing strategy whichforecasts the active bounds at a stationary point, the determination of a descent direction bymeans of solving a reduced strictly convex quadratic program with box constraints and anexact line search. Global convergence is established in the sense that every accumulationpoint is stationary. Moreover, it is shown that the algorithm terminates after a finite numberof iterations, if at least one iterate is sufficiently close to a stationary point which satisfiesa certain sufficient optimality condition. The algorithm can be easily implemented for sparselarge-scale BQPs. Furthermore, it simplifies for concave BQPs, as it is not required to solvestrictly convex quadratic programs in this case. Computational experience with large-scaleBQPs is included and shows the appropriateness of this type of methodology

The eigenvalue complementarity problem

Abstract In this paper an eigenvalue complementarity problem (EiCP) is studied, which finds its origins in the solution of a contact problem in mechanics. The EiCP is shown to be equivalent to a Nonlinear Complementarity Problem, a Mathematical Programming Problem with Complementarity Constraints and a Global Optimization Problem. A finite Reformulation–Linearization Technique (Rlt)-based tree search algorithm is introduced for processing the EiCP via the lattermost of these formulations. Computational experience is included to highlight the efficacy of the above formulations and corresponding techniques for the solution of the EiCP