This paper deals with a problem from discrete-time robust control which requires the solution of constraints over the reals that contain both universal and existential quantifiers. For solving this problem we formulate it as a program in a (fictitious) constraint logic programming language with explicit quantifier notation. This allows us to clarify the special structure of the problem, and to extend an algorithm for computing approximate solution sets of first-order constraints over the reals to exploit this structure. As a result we can deal with inputs that are clearly out of reach for current symbolic solvers.
Apart from the method used in this paper @cite_8 , there there have been several successful attempts at solving special cases of first-order constraints, for example using classical interval techniques @cite_36 @cite_21 or constraint satisfaction @cite_0 , and very often in the context of robust control @cite_2 @cite_30 @cite_28 @cite_11 .
@cite_30 @cite_8 @cite_36 @cite_28 @cite_21 @cite_0 @cite_2 @cite_11
2768546550 2018738327 2126442135 1982831910
First-order methods have been popularly used for solving large-scale problems. However, many existing works only consider unconstrained problems or those with simple constraint. In this paper, we develop two first-order methods for constrained convex programs, for which the constraint set is represented by affine equations and smooth nonlinear inequalities. Both methods are based on the classic augmented Lagrangian function. They update the multipliers in the same way as the augmented Lagrangian method (ALM) but employ different primal variable updates. The first method, at each iteration, performs a single proximal gradient step to the primal variable, and the second method is a block update version of the first one. For the first method, we establish its global iterate convergence as well as global sublinear and local linear convergence, and for the second method, we show a global sublinear convergence result in expectation. Numerical experiments are carried out on the basis pursuit denoising and a convex quadratically constrained quadratic program to show the empirical performance of the proposed methods. Their numerical behaviors closely match the established theoretical results. When solving the general smooth nonlinear and possibly nonconvex optimization problem involving equality and or inequality constraints, an approximate first-order critical point of accuracy @math can be obtained by a second-order method using cubic regularization in at most @math evaluations of problem functions, the same order bound as in the unconstrained case. This result is obtained by first showing that the same result holds for inequality constrained nonlinear least-squares. As a consequence, the presence of (possibly nonconvex) equality inequality constraints does not affect the complexity of finding approximate first-order critical points in nonconvex optimization. This result improves on the best known ( @math ) evaluation-complexity bound for solving general nonconvexly constrained optimization problems. We present a simple probabilistic algorithm for solving k-SAT and more generally, for solving constraint satisfaction problems (CSP). The algorithm follows a simple local search paradigm (S. , 1992): randomly guess an initial assignment and then, guided by those clauses (constraints) that are not satisfied, by successively choosing a random literal from such a clause and flipping the corresponding bit, try to find a satisfying assignment. If no satisfying assignment is found after O(n) steps, start over again. Our analysis shows that for any satisfiable k-CNF-formula with n variables this process has to be repeated only t times, on the average, to find a satisfying assignment, where t is within a polynomial factor of (2(1-1 k)) sup n . This is the fastest (and also the simplest) algorithm for 3-SAT known up to date. We consider also the more general case of a CSP with n variables, each variable taking at most d values, and constraints of order l, and analyze the complexity of the corresponding (generalized) algorith m. It turns out that any CSP can be solved with complexity at most (d spl middot (1-1 l)+ spl epsiv ) sup n . In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order @math , instead of the usual @math Gauss--Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20 faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks.