Sequence Of Stationary Points Research Articles

A Quantum Model of the Distribution of Prime Numbers and the Riemann Hypothesis

After proving that numerical sequences such as Fibonacci numbers and prime numbers, can be generated as sequences of equilibrium points of an ideal half-infinite one-dimensional distribution of electric charges, a model of the distribution of primes on the x-axis is proposed, where primes ρ(n) are considered as quantum particles oscillating around the sequence of stationary points r(n) of the Lennard-Jones-like potential of the single-particle Hamiltonian. A particle-counting function πQ(x) is defined over the many-particle system, in the same way as the prime-counting function π(x). Through the application of the Hellmann-Feynman theorem, the existence of a solution n(x) ≈ li(x), coinciding asymptotically with the logarithmic integral function, is proved for the counting function of stationary points, together with the quantum equivalent of the Prime Number Theorem for the particle-counting function πQ(x). The conditions on the sequence of energy eigenvalues E(n) of the system, so that the Riemann hypothesis holds for the function πQ(x), are derived, thus proving the quantum theory results in the same bound of the position of the prime-particle in the n-th quantum state as that derived through analytical methods for the n-th prime. The model suggests new conjectures and statistical procedures which are applied to study the sign of the difference π(x)-li(x) and to explore the region beyond the Skewes number. The data predicted by the theory find confirmation when compared with those known in the literature. A new statistical local test of the Riemann hypothesis, based on the difference π(x)-li(x), is proposed.

On Approximate Stationary Points of the Regularized Mathematical Program with Complementarity Constraints

We discuss the convergence of regularization methods for mathematical programs with complementarity constraints with approximate sequence of stationary points. It is now well accepted in the literature that, under some tailored constraint qualification, the genuine necessary optimality condition for this problem is the M-stationarity condition. It has been pointed out, (Kanzow and Schwartz in Math Oper Res 40(2):253–275. 2015), that relaxation methods with approximate stationary points fail to ensure convergence to M-stationary points. We define a new strong approximate stationarity concept, and we prove that a sequence of strong approximate stationary points always converges to an M-stationary solution. We also prove under weak assumptions the existence of strong approximate stationary points in the neighborhood of an M-stationary solution.

A sample average approximation regularization method for a stochastic mathematical program with general vertical complementarity constraints

Based on the log-exponential function, a sample average approximation (SAA) regularization method is proposed for solving a stochastic mathematical program with general vertical complementarity constraints (SMPVCC) considered by Birbil et al. (2006). Detailed convergence analysis of this method is investigated. It is demonstrated that under some regularity conditions, any accumulation point of the sequence of optimal solutions of SAA regularized problem is almost surely an optimal solution of the SMPVCC as the parameter tends to zero and the sample size tends to infinity. Furthermore, the optimal value sequence of SAA regularized problem converges to the optimal value of SMPVCC with exponential convergence rate with probability one and a sequence of stationary points of regularized SAA problem converges almost surely to a stationary point of SMPVCC. Finally, we show that a stochastic Stackelberg game can be formulated as a SMPVCC problem and an equilibrium solution can be obtained by the method proposed.

Notes on Convergence Properties for a Smoothing-Regularization Approach to Mathematical Programs with Vanishing Constraints

We give some improved convergence results about the smoothing-regularization approach to mathematical programs with vanishing constraints (MPVC for short), which is proposed in Achtziger et al. (2013). We show that the Mangasarian-Fromovitz constraints qualification for the smoothing-regularization problem still holds under the VC-MFCQ (see Definition 5) which is weaker than the VC-LICQ (see Definition 7) and the condition of asymptotic nondegeneracy. We also analyze the convergence behavior of the smoothing-regularization method and prove that any accumulation point of a sequence of stationary points for the smoothing-regularization problem is still strongly-stationary under the VC-MFCQ and the condition of asymptotic nondegeneracy.

Convergence Properties of a Smoothing Approach for Mathematical Programs with Second-Order Cone Complementarity Constraints

This paper discusses the convergence properties of a smoothing approach for solving the mathematical programs with second-order cone complementarity constraints (SOCMPCCs). We first introduce B-stationary, C-stationary, M(orduckhovich)-stationary, S-stationary point, SOCMPCC-linear independence constraint qualification (denoted by SOCMPCC-LICQ), second-order cone upper level strict complementarity (denoted by SOC-ULSC) condition at a feasible point of a SOCMPCC problem. With the help of the projection operator over a second-order cone, we construct a smooth optimization problem to approximate the SOCMPCC. We demonstrate that any accumulation point of the sequence of stationary points to the sequence of smoothing problems, when smoothing parameters decrease to zero, is a C-stationary point to the SOCMPCC under SOCMPCC-LICQ at the accumulation point. We also prove that the accumulation point is an M-stationary point if, in addition, the sequence of stationary points satisfy weak second order necessary conditions for the sequence of smoothing problems, and moreover it is a B-stationary point if, in addition, the SOC-ULSC condition holds at the accumulation point.

A Smoothing Method for Zero–One Constrained Extremum Problems

In this paper, the zero–one constrained extremum problem is reformulated as an equivalent smooth mathematical program with complementarity constraints (MPCC), and then as a smooth ordinary nonlinear programming problem with the help of the Fischer–Burmeister function. The augmented Lagrangian method is adopted to solve the resulting problem, during which the non-smoothness may be introduced as a consequence of the possible inequality constraints. This paper incorporates the aggregate constraint method to construct a uniform smooth approximation to the original constraint set, with approximation controlled by only one parameter. Convergence results are established, showing that under reasonable conditions the limit point of the sequence of stationary points generated by the algorithm is a strongly stationary point of the original problem and satisfies the second order necessary conditions of the original problem. Unlike other penalty type methods for MPCC, the proposed algorithm can guarantee that the limit point of the sequence is feasible to the original problem.

Smoothing method for mathematical programs with symmetric cone complementarity constraints

In this article, we consider the mathematical program with symmetric cone complementarity constraints (MPSCCC) in a general form. It contains the mathematical program with second-order-cone complementarity constraints and the mathematical program with complementarity constraints. We present a smoothing method that approximates the primal MPSCCC by means of the Chen–Mangasarian class of smoothing functions. We show that a sequence of stationary points of the approximate programs converges to a C(larke)-stationary point of the primal MPSCCC under suitable conditions.

On the convergence of general regularization and smoothing schemes for mathematical programs with complementarity constraints

We extend the convergence analysis of a smoothing method [M. Fukushima and J.-S. Pang (2000). Convergence of a smoothing continuation method for mathematical programs with complementarity constraints. In: M. Théra and R. Tichatschke (Eds.), Ill-posed Variational Problems and Regularization Techniques, pp. 99–110. Springer, Berlin/Heidelberg.] to a general class of smoothing functions and show that a weak second-order necessary optimality condition holds at the limit point of a sequence of stationary points found by the smoothing method. We also show that convergence and stability results in [S. Scholtes (2001). Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim., 11, 918–936.] hold for a relaxation problem suggested by Scholtes [S. Scholtes (2003). Private communications.] using a class of smoothing functions. In addition, the relationship between two technical, yet critical, concepts in [M. Fukushima and J.-S. Pang (2000). Convergence of a smoothing continuation method for mathematical programs with complementarity constraints. In: M. Théra and R. Tichatschke (Eds.), Ill-posed Variational Problems and Regularization Techniques, pp. 99–110. Springer, Berlin/Heidelberg; S. Scholtes (2001). Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim., 11, 918–936.] for the convergence analysis of the smoothing and regularization methods is discussed and a counter-example is provided to show that the stability result in [S. Scholtes (2001). Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim., 11, 918–936.] cannot be extended to a weaker regularization.