In this paper, we present an efficient QR algorithm for solving the linear response eigenvalue problem H x= x, where H is -symmetric with respect to 0= diag (I n, I n). Based on newly introduced -orthogonal transformations, the QR algorithm preserves the -symmetric structure of H throughout the whole process, and thus guarantees the computed eigenvalues to appear pairwise (, ) as they should. With the help of a newly established implicit -orthogonality theorem, we incorporate the implicit multi-shift technique to accelerate the convergence of the QR algorithm. Numerical experiments are given to show the effectiveness of the algorithm.

We observe that Sturm’s error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of O(k^{-1/(2^{d+1}-2)}), 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 O(1/k^{0.5}).

Stochastic gradient (SG) method has been popularly applied to solve optimization problems with objective that is stochastic or an average of many functions. Most existing works on SG assume that the underlying problem is unconstrained or has an easy-to-project constraint set. In this paper, we consider problems that have a stochastic objective and also many functional constraints. For such problems, it could be extremely expensive to project a point to the feasible set, or even compute subgradient and/or function value of all constraint functions. To find solutions of these problems, we propose a novel SG method based on the augmented Lagrangian function. Within every iteration, it inquires a stochastic subgradient of the objective, a subgradient and function value of one randomly sampled constraint function, and function value of another sampled constraint function. Hence, the per-iteration complexity is low. We establish its convergence rate for convex and also strongly convex problems. It can achieve the optimal O (1/\sqrt {k}) convergence rate for convex case and nearly optimal O (1/\sqrt {k}) rate for strongly convex case. Numerical experiments on quadratically constrained quadratic programming are conducted to demonstrate its efficiency.

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Block coordinate update (BCU) methods enjoy low per-update computational complexity because every time only one or a few block variables would need to be updated among possibly a large number of blocks. They are also easily parallelized and thus have been particularly popular for solving problems involving large-scale dataset and/or variables. In this paper, we propose a primaldual BCU method for solving linearly constrained convex program with multi-block variables. The method is an accelerated version of a primaldual algorithm proposed by the authors, which applies randomization in selecting block variables to update and establishes an <i>O</i>(1/<i>t</i>) convergence rate under convexity assumption. We show that the rate can be accelerated to O ( 1 / t 2 ) if the objective is strongly convex. In addition, if one block variable is independent of the others in the objective, we then show that the algorithm can

We propose an algorithm for the state feedback pole assignment problem. The algorithm is the first of its kind, making use of the Schur form and minimizing the departure from normality of the closed-loop poles by Newtons method. Eleven illustrative examples, comparing our algorithm with three other existing ones, are given.

We discuss a variant of the multi-task <i>n</i>-vehicle exploration problem. Instead of requiring an optimal permutation of vehicles in every group, the new problem requires all vehicles in a group to arrive at the same destination. Given <i>n</i> tasks with assigned consume-time and profit, it may also be viewed as a maximization of every processors average profit. Further, we propose a new kind of partition problem in fractional form and analyze its computational complexity. By regarding fractional partition as a special case, we prove that the average profit maximization problem is NP-hard when the number of processors is fixed and it is strongly NPhard in general. At last, a pseudo-polynomial time algorithm for the average profit maximization problem and the fractional partition problem is presented, using the idea of the pseudo-polynomial time algorithm for the classical partition problem.

Recent several years have witnessed the surge of asynchronous (async-) parallel computing methods due to the extremely big data involved in many modern applications and also the advancement of multi-core machines and computer clusters. In optimization, most works about async-parallel methods are on unconstrained problems or those with block separable constraints. In this paper, we propose an async-parallel method based on block coordinate update (BCU) for solving convex problems with <i>nonseparable</i> linear constraint. Running on a single node, the method becomes a novel randomized primaldual BCU for multi-block affinely constrained problems. For these problems, GaussSeidel cyclic primaldual BCU is not guaranteed to converge to an optimal solution if no additional assumptions, such as strong convexity, are made. On the contrary, assuming convexity and existence of a primaldual