First and second order analysis of nonlinear semidefinite programs

In this paper we study nonlinear semidefinite programming problems. Convexity, duality and first-order optimality conditions for such problems are presented. A second-order analysis is also given. Second-order necessary and sufficient optimality conditions are derived. Finally, sensitivity analysis of such programs is discussed.     

A smoothing method for second order cone complementarity problem

In this paper, the second order cone complementarity problem is studied. Based on a perturbed symmetrically smoothing function, which has coerciveness under proper conditions, we present a smoothing Newton method for this problem. The boundedness of the level set can be obtained from the coerciveness, which plays an important role in the convergence analysis. Furthermore, the proposed algorithm for the reformulation has no restrictions on the starting point and solves only one system of equations. Preliminary numerical results indicate that the algorithm is effective.     

A note on temporal regularity of second order difference equations

This work deals with the study of maximal ?^p‐regularity of a pair (A,B) of bounded linear operators on a complex Banach space associated to the second order difference equation u_{n + 2} = Bu_{n + 1} + Au_n + f_n, where f is a given sequence on . We obtain results of characterization based on spectral analysis of the discrete sine family, which is the resolvent family of this equation.     

A note on a special case of the Frobenius problem

For a set of positive and relative prime integers A = {a ₁…,a _k }, let Γ(A) denote the set of integers of the form a ₁ x ₁+…+a _k x _k with each x _j ≥ 0. Let g(A) (respectively, n(A) and s(A)) denote the largest integer (respectively, the number of integers and sum of integers) not in Γ(A). Let S*(A) denote the set of all positive integers n not in Γ(A) such that n + Γ(A) \ {0} ? Γ((A)\{0}. We determine g(A), n(A), s(A), and S*(A) when A = {a, b, c} with a | (b + c).     

A note on homoclinic orbits for second order Hamiltonian system

Some existence and multiplicity of homoelinic orbits for second order Hamiltonian system x-a(t)x f(t,x)=0 are given by means of variational methods, where the function -1/2a（t）|s|^2∫^t0f（t,s）ds is asymptotically quadratic in s at infinity and subquadratic in s at zero, and the function a (t) mainly satisfies the growth condition limt→∞∫^t 1 t a（t）dt= ∞,VI∈R^1.A resonance case as well as a noncompact case is discussed too.     

A note on the positive solutions of second order differential equations

We obtain an existence theorem for monotone positive solutions of nonlinear second-order nonlinear differential equations via principal and nonprincipal solutions of the corresponding homogeneous equation.     

A class of nonlinear Lagrangians for nonconvex second order cone programming

This paper focuses on the study of a class of nonlinear Lagrangians for solving nonconvex second order cone programming problems. The nonlinear Lagrangians are generated by Löwner operators associated with convex real-valued functions. A set of conditions on the convex real-valued functions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms. These conditions are satisfied by well-known nonlinear Lagrangians appeared in the literature. The convergence properties for the nonlinear Lagrange method are discussed when subproblems are assumed to be solved exactly and inexactly, respectively. The convergence theorems show that, under the second order sufficient conditions with sigma-term and the strict constraint nondegeneracy condition, the algorithm based on any of nonlinear Lagrangians in the class is locally convergent when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Compared to the analysis in nonlinear Lagrangian methods for nonlinear programming, we have to deal with the sigma term in the convergence analysis. Finally, we report numerical results by using modified Frisch’s function, modified Carroll’s function and the Log-Sigmoid function.     

A special kind of nonoscillatory second order linear differential equations

In this note the nonoscillatory property ofy +p (x)y = 0, wherep (x) is a periodic pulse function, is considered. Sufficient condition for guaranteeing nonscillation is obtained. We compare the result with obtained by applying Adamov's theorem.     

A note on Kamenev type theorems for second order matrix differential systems

Some oscillation criteria are given for the second order matrix differential system , where and are real continuous matrix functions with symmetric, . These results improve oscillation criteria recently discovered by Erbe, Kong and Ruan by using a generalized Riccati transformation , where is the identity matrix, is a given function on and .

     

关于半定规划的一种宽邻域不可行内点算法的注记

针对半定规划的宽邻域不可行内点算法, 将牛顿法和预估校正法进行结合, 构造出适当的迭代方向, 提出一个修正的半定规划宽邻域不可行内点算法, 并在适当的假设条件下, 证明了该算法具有O(\sqrt{n}L)的迭代复杂界．最后利用Matlab编程, 给出了基于KM方向和NT方向的数值实验结果．     

How to convexify the intersection of a second order cone and a nonconvex quadratic

A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown—by several authors using different techniques—that the convex hull of the intersection of an ellipsoid, \(\mathcal {E}\), and a split disjunction, \((l - x_j)(x_j - u) \le 0\) with \(l < u\), equals the intersection of \(\mathcal {E}\) with an additional second-order-cone representable (SOCr) set. In this paper, we study more general intersections of the form \(\mathcal {K}\cap \mathcal {Q}\) and \(\mathcal {K}\cap \mathcal {Q}\cap H\), where \(\mathcal {K}\) is a SOCr cone, \(\mathcal {Q}\) is a nonconvex cone defined by a single homogeneous quadratic, and H is an affine hyperplane. Under several easy-to-verify conditions, we derive simple, computable convex relaxations \(\mathcal {K}\cap \mathcal {S}\) and \(\mathcal {K}\cap \mathcal {S}\cap H\), where \(\mathcal {S}\) is a SOCr cone. Under further conditions, we prove that these two sets capture precisely the corresponding conic/convex hulls. Our approach unifies and extends previous results, and we illustrate its applicability and generality with many examples.     

A note on the existence of multiple solutions for a class of systems of second order ODEs

We prove the existence of multiple solutions for some systems of second order ODEs with Dirichlet boundary conditions. Such systems are obtained by coupling scalar ODEs with different growth conditions. The proof relies on a global continuation technique.     

A note on alternating projections for ill-posed semidefinite feasibility problems

We observe that Sturm’s error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of \(\mathcal {O}\Big (k^{-\frac{1}{2^{d+1}-2}}\Big )\), where d is the singularity degree of the problem—the minimal number of facial reduction iterations needed to induce Slater’s condition. Consequently, for almost all such problems (in the sense of Lebesgue measure), alternating projections converge at a worst-case rate of \(\mathcal {O}\Big (\frac{1}{\sqrt{k}}\Big )\).     

A note on sturmian comparison theorems for linear partial differential equations of second order

We show that by a modification of Sturm's classical method it is possible to obtain results for special operators of mixed type as well as for nonhomogeneous equations     

A note on semidefinite programming relaxations for polynomial optimization over a single sphere

We study two instances of polynomial optimization problem over a single sphere. The first problem is to compute the best rank-1 tensor approximation. We show the equivalence between two recent semidefinite relaxations methods. The other one arises from Bose-Einstein condensates (BEC), whose objective function is a summation of a probably nonconvex quadratic function and a quartic term. These two polynomial optimization problems are closely connected since the BEC problem can be viewed as a structured fourth-order best rank-1 tensor approximation. We show that the BEC problem is NP-hard and propose a semidefinite relaxation with both deterministic and randomized rounding procedures. Explicit approximation ratios for these rounding procedures are presented. The performance of these semidefinite relaxations are illustrated on a few preliminary numerical experiments.     

A complete characterization of disjunctive conic cuts for mixed integer second order cone optimization

We study the convex hull of the intersection of a disjunctive set defined by parallel hyperplanes and the feasible set of a mixed integer second order cone optimization (MISOCO) problem. We extend our prior work on disjunctive conic cuts (DCCs), which has thus far been restricted to the case in which the intersection of the hyperplanes and the feasible set is bounded. Using a similar technique, we show that one can extend our previous results to the case in which that intersection is unbounded. We provide a complete characterization in closed form of the conic inequalities required to describe the convex hull when the hyperplanes defining the disjunction are parallel.