# LINEAR COMPLEMENTARITY PROBLEMS

## Top PDF LINEAR COMPLEMENTARITY PROBLEMS:

### Infeasible path following algorithms for linear complementarity problems

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### Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix

there are doubly active indices, the Jacobian used in ( 1.3 ), determined by the choice ( 1.2 ), is an element of the Clarke generalized Jacobian [ 4 ] of the function in ( 1.4 ). This description makes it natural to call Newton-min the algorithm that updates x by the formulas ( 1.2 )-( 1.3 ). The algorithm sketched above and that we further explore in this paper can be traced back at least to the work of Aganagi´c [ 1 , 1984], who proposed a Newton-type algorithm for solving LC(M, q) when M is a hidden Z-matrix, which is a matrix that can be written M X = Y where X and Y are Z-matrices having particular properties (an M -matrix is a particular instance of hidden Z-matrix). In his approach, the nonsmooth equation ( 1.4 ) to solve is then expressed in terms of a Z-matrix associated with M ; a convergence result is established. The algorithm was then rediscovered in the form ( 1.2 )–( 1.3 ) by Bergounioux, Ito, and Kunisch [ 2 , 1999] for solving quadratic optimal control problems under the name of primal-dual active set strategy (see also [ 12 , 2008]). Hinterm¨ uller [ 10 , 2003] uses the same algorithm for solving constrained optimal control problems, shows that the method does not cycle in the case of bilateral constraints, and proves convergence in finite and infinite dimension. It is shown in [ 11 , 2003] that the algorithm can be viewed as a nonsmooth Newton method for solving ( 1.4 ), which motivates the name of the algorithm given in this note, and that it converges locally superlinearly if M is a P -matrix. Kanzow [ 13 , 2004] proved its convergence in at most n iterations when M is an M -matrix.
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### A relation between homotopy and pivotal methods for linear complementarity problems

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### Time-Stepping via Complementarity

The last approaches are based on the non-smooth modelling and simulation framework. The two last approaches only differ in their time-integration techniques. For the modelling part, the goal is to design some ideal electrical models by means of formulations extensively used in the mathematical programming theory. To list a few of them, the models that are based on variational inequalities (VI) [ 24 ], com- plementarity problems (CP) [ 22 ], inclusions into a normal cone to a convex set [ 46 ] and non-smooth and convex optimisation problems [ 33 ] are very good candidates. The main difference with the hybrid approach is the intrinsic functional description (possibly multi-valued) of the component rather than a discrete/modal description. In Fig. 14.1 e, a complementarity modelling of the diode with possible residual cur- rent b and voltage a is depicted. It can be defined by the following complementarity condition, or inclusion into a normal cone, as
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### Linear inverse problems with noise: primal and primal-dual splitting

jean-luc.starck@cea.fr ABSTRACT In this paper, we propose two algorithms for solving lin- ear inverse problems when the observations are corrupted by noise. A proper data fidelity term (log-likelihood) is in- troduced to reflect the statistics of the noise (e.g. Gaus- sian, Poisson). On the other hand, as a prior, the images to restore are assumed to be positive and sparsely repre- sented in a dictionary of waveforms. Piecing together the data fidelity and the prior terms, the solution to the in- verse problem is cast as the minimization of a non-smooth convex functional. We establish the well-posedness of the optimization problem, characterize the corresponding min- imizers, and solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory. Experimental re- sults on deconvolution, inpainting and denoising with some comparison to prior methods are also reported.
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### Domain decomposition methods for non linear problems in fluid dynamics

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### Application of linear routing systems to regional groundwater problems.

NO,NRES NO (Col. 1 represents the Linear Reservoir model, 2 the Lag and Route model, and 3 the Nash model as discussed in Chapter III. 4-13) indicates the number of equal linear rese[r]

### An automated tool for formulating linear covariance analysis problems

The only non- zero elements are those corresponding to the sensor taking the measurement, its host platform position and velocity error vectors, and the position and veloc[r]

### Bifurcation and non linear eigenvalue problems: a tentative bibliography

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### Linear boundary-value problems described by Drazin invertible operators

References J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal. 243 (2007), 536-565. P.L. Butzer, J.J. Koliha, The a-Drazin inverse and ergodic behaviour of semigroups and cosine operator functions, J. Operator theory 62; 2(2009), 297-326. N. Khaldi, M. Benharrat, B. Messirdi, On the Spectral Boundary Value Problems and Boundary Approximate Controllability of Linear Systems. Rend. Circ. Mat. Palermo, 63 (2014) 141-153.

### Solution of Non-Linear Elasticity Problems Using the Continu Software

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### On the Complexity of Linear and Stratified : Context Matching Problems

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### Asymetric quasi linear finite element methods for solving non-linear incompressible elasticity problems

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### The convex algebraic geometry of linear inverse problems

problem subject to linear constraints. Recovering measures Recovering a measure given its moments is another question of interest. Typically one is given some set of moments up to order d, and the goal is to recover an atomic measure that is consistent with these moments. In the limited information setting, we wish to recover a measure given a small set of linear combinations of the moments. Such a question is also of interest in system identification. The set A in this setting is the moment curve, and its convex hull goes by several names including the Caratheodory orbitope [18]. Discretized versions of this problem correspond to the set A being a finite number of points on the moment curve; the convex hull C( A ) is then a cyclic polytope [19].
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### The application of linear operators to geophysical problems

The two-dinsional operator is illustrated by deriving an operator suitable for the detection of an anomaly in the presence of noise, in the form of a regional gr[r]

### Several Graph problems and their Linear Program formulations

5 Linear arboricity The linear arboricity of a graph G is the least number k such that the edges of G can be partitionned into k classes, each of them being a forest of paths (the disjoints union of paths – trees of maximal degree 2). The corresponding LP is very similar to the one giving edge-disjoint spanning trees

### Efficient computation of condition estimates for linear least squares problems

of LLS problems, using the Frobenius norm to measure the perturbations of A. Since then many results on normwise LLS condition numbers have been published (see e.g. [2, 6, 11, 15, 16]). It was observed in [18] that normwise condition numbers can lead to a loss of information since they consolidate all sensitivity information into a single number. Indeed in some cases this sensitivity can vary significantly among the different solution components (some examples for LLS are presented in [2, 22]). To overcome this issue, it was proposed the notion of “componentwise” condition numbers or condition numbers for the solution components [9]. Note that this approach must be distinguished from the componentwise metric also applied to LLS for instance in [4, 10]. This approach was generalized by the notion of partial or subspace condition numbers where we study the conditioning of L T x with L ∈ R n×k , k
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### Measuring complementarity in financial systems

Thakor (1997) show using a theoreti al model that by oordinating the agents' a tions, banks resolve asset substitution problems, whereas nan ial markets improve the real de ision of a rm due to feed ba k from market pri es. The on lusions of these studies suggest that bank and market based systems dier in their performan e, and it is important to understand these

### New quadratic solid-shell elements and their evaluation on linear benchmark problems

This last benchmark problem also deals with linear elastic stability. The corresponding buckling analysis is based on the eigenvalue equation ( 12 ), which allows the formula- tion of the geometric stiffness matrix K σ to be checked once again. The test consists of a portion of a submarine hull subjected to external pressure, as illustrated in Fig. 6 . This test was previously used in [ 40 ], while in [ 41 ] the submarine was modeled in its entirety. The current analysis only considers a single ring, as described in Fig. 6 , and the geometric and material data are reported in Table 9 .

### On the parallel scalability of hybrid linear solvers for large 3D problems

Figure 3.2: Performance comparison of dense v.s. sparse Cholesky factorization. 3.4 Mixed precision Additive Schwarz preconditioner Motivated by accuracy reasons, many large-scale scientific applications and industrial numerical simulation codes are fully implemented in 64-bit floating-point arithmetic. On the other hand, many recent processor architectures exhibit 32-bit computational power that is significantly higher than that for 64-bit. One recent and significant example is the IBM CELL multiprocessor that is projected to have a peak performance near 256 GFlops in 32-bit and “only” 26 GFlops in 64-bit computation. More extreme and common examples are the processors that possess a SSE (streaming SIMD exten- sion) execution unit can perform either two 64-bit instructions or four 32-bit instructions in the same time. This class of chip includes for instance the IBM PowerPC G5, the AMD Opteron and the Intel Pentium. For illustration purpose, are displayed below the time and the ratio of the time to perform a 32-bit operation over the time to perform the corresponding 64-bit one on some of the basic dense kernels involved in our hybrid solver implementation. In Table 3.4 are displayed the performance of B LAS -2 (_GEMV) and B LAS -3 (_GEMM) routines for various problems sizes. The comparison
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