An extension lemma and homogeneous programming

In existence theorems for optimal solutions in control theory, concepts of upper semicontinuity of variable sets in Euclidean spaces have been used. In the present paper, a scale of concepts is introduced which is intermediate between Kuratovski's upper semicontinuity and Cesari's upper semicontinuity. First, the relationships and properties of the concepts are studied. Then, examples of problems of control with distributed and boundary parameters are given showing the relevance of the same concepts.

The optimality conditions of major constraints nonsmooth programming with multi-objectives are studied. The objective function of this class of multi-objective programming is the sum of a convex vector function and a differentiable vector function. The constraint condition is major cone constraint in Euclid space. By using of the structure representation of major cone constraint of the given problem, the Fritz John condition of Pareto weakly effective solution is obtained by Gordan theorem. Meanwhile, the Kuhn-Tucker condition for Pareto weakly effective solution under Slater constraint qualification is given. The results are very useful to design its numerical methods.

In this article, PAC-learning theory is applied to model inference, which concerns the problem of inferring theories from facts in first order logic. It is argued that uniform sample PAC-learnability cannot be expected with most of the ‘interesting’ model classes. Polynomial sample learnability can only be accomplished in classes of programs having a fixed maximum number of clauses. We have proved that the class of context free programs in a fixed maximum number of clauses with a fixed maximum number of literals is learnable from a polynomial number of examples. This is also proved for a more general class of programs.

Total unimodularity applications in bi-objective discrete optimization

The paper presents a method for fast real-time solution of a class of multi-parametric mixed integer quadratic programs. The complexity of the problem solved in real time is significantly reduced by moving part of the computational burden off-line. In the off-line computations, the structure of the mixed integer program is explored to determine a polytopic partition of the parameter set. Each polytope is then considered to determine which integer combinations can/cannot be optimal, given that the parameter is contained in the current polytope. The real-time solution of the mixed integer program consists of a two step procedure: First it is determined which polytope contains the parameter. Then the mixed integer quadratic program is solved, taking into account only those integer combinations associated with the current polytope. If the number of integer combinations is small, the overall complexity of the real-time solution procedure will be reduced.

We consider the solution of convex programs in a small number of variables but large number of constraints, where all but a small number of the constraints are linear. We develop a general framework for obtaining algorithms for these problems which run in time linear in the number of constraints. We give an application to computing minimum spanning ellipsoids in fixed dimension.

Mathematical programs with cardinality constraints are optimization problems with an additional constraint which requires the solution to be sparse in the sense that the number of nonzero elements, i.e. the cardinality, is bounded by a given constant. Such programs can be reformulated as a mixed-integer ones in which the sparsity is modeled with the use of complementarity-type constraints. It is shown that the standard relaxation of the integrality leads to a nonlinear optimization program of the striking property that its solutions (global minimizers) are the same as the solutions of the original program with cardinality constraints. Since the number of local minimizers of the relaxed program is typically larger than the number of local minimizers of the cardinality-constrained problem, the relationship between the local minimizers is also discussed in detail. Furthermore, we show under which assumptions the standard KKT conditions are necessary optimality conditions for the relaxed program. The main result obtained for such conditions is significantly different from the existing optimality conditions that are known for the somewhat related class of mathematical programs with complementarity constraints.

We address the exact solution of general integer quadratic programs with linear constraints. These programs constitute a particular case of mixed-integer quadratic programs for which we introduce in Billionnet et al. (Math. Program., 2010) a general solution method based on quadratic convex reformulation, that we called MIQCR. This reformulation consists in designing an equivalent quadratic program with a convex objective function. The problem reformulated by MIQCR has a relatively important size that penalizes its solution time. In this paper, we propose a convex reformulation less general than MIQCR because it is limited to the general integer case, but that has a significantly smaller size. We call this approach Compact Quadratic Convex Reformulation (CQCR). We evaluate CQCR from the computational point of view. We perform our experiments on instances of general integer quadratic programs with one equality constraint. We show that CQCR is much faster than MIQCR and than the general non-linear solver BARON (Sahinidis and Tawarmalani, User's manual, 2010) to solve these instances. Then, we consider the particular class of binary quadratic programs. We compare MIQCR and CQCR on instances of the Constrained Task Assignment Problem. These experiments show that CQCR can solve instances that MIQCR and other existing methods fail to solve.

Solving linear programs is often a challenging task in distributed settings. While there are good algorithms for solving packing and covering linear programs in a distributed manner (Kuhn et al. 2006), this is essentially the only class of linear programs for which such an algorithm is known. In this work we provide a distributed algorithm for solving a different class of convex programs which we call “distance-bounded network design convex programs”. These can be thought of as relaxations of network design problems in which the connectivity requirement includes a distance constraint (most notably, graph spanners). Our algorithm runs in O((D/e) log n) rounds in the LOCAL model and with high probability finds a (1+e)-approximation to the optimal LP solution for any 0 < e ≤ 1, where D is the largest distance constraint. While solving linear programs in a distributed setting is interesting in its own right, this class of convex programs is particularly important because solving them is often a crucial step when designing approximation algorithms. Hence we almost immediately obtain new and improved distributed approximation algorithms for a variety of network design problems, including Basic 3- and 4-Spanner, Directed k-Spanner, Lowest Degree k-Spanner, and Shallow-Light Steiner Network Design with a spanning demand graph. Our algorithms do not require any “heavy” computation and essentially match the best-known centralized approximation algorithms, while previous approaches which do not use heavy computation give approximations which are worse than the best-known centralized bounds.

We present a probabilistic logic programming framework that allows the representation of conditional probabilities. While conditional probabilities are the most commonly used method for representing uncertainty in probabilistic expert systems, they have been largely neglected by work in quantitative logic programming. We define a fixpoint theory, declarative semantics, and proof procedure for the new class of probabilistic logic programs. Compared to other approaches to quantitative logic programming, we provide a true probabilistic framework with potential applications in probabilistic expert systems and decision support systems. We also discuss the relationship between such programs and Bayesian networks, thus moving toward a unification of two major approaches to automated reasoning.

A class of nonsmooth multiobjective fractional programming is formulated. We establish the necessary and sufficient optimality conditions without the need of a constraint qualification. Then a mixed dual is introduced for a class of nonsmooth fractional programming problems, and various duality theorems are established without a constraint qualification.

In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a (T+1)-stage stochastic MINLP satisfying L-exact Lipschitz regularization with d-dimensional state spaces, to obtain an varepsilon -optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by {mathcal {O}}((frac{2LT}{varepsilon })^d), and is lower bounded by {mathcal {O}}((frac{LT}{4varepsilon })^d) for the general case or by {mathcal {O}}((frac{LT}{8varepsilon })^{d/2-1}) for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity depends polynomially on the number of stages. We further show that the iteration complexity depends linearly on T, if all the state spaces are finite sets, or if we seek a (Tvarepsilon )-optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale with T. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization.

The PPL framework is proposed as a simple extension to logic programming aiming at handling resources. It is argued that the separation between logical treatments and resource handling is desirable and, to that end, resources are proposed to be manipulated by means of pre- and post-conditions associated with usual Horn clauses. The expressiveness of the resulting framework is evidenced through the coding of several applications involving objects, databases, actions and changes. Operational and declarative semantics are presented as well. The operational semantics rests on a derivation relation stating how goals and conditions are evaluated. The declarative semantics extends the classical model and fixed-point theories to take into account the evaluation of pre- and post-conditions, and in particular the non-monotonic behavior of the world of resources they induce in general. As suggested, an effort has been made to keep the work close to the classical logic programming setting. In particular, the semantics are in the main streams of logic programming semantics. However, the PPL framework raises new problems for which fresh solutions are proposed.

Fuzzy logic is a topic traditionally taught in artificial intelligence, machine learning, and robotics courses. Students receive the necessary mathematical and theoretical foundation in lecture format. The final learning experience may require that students create and code their own fuzzy logic application that solves a real world problem. This can be an issue when the target is a bioengineering course that introduces classical control theory, fuzzy logic, neural networks, genetic algorithms and genetic programming through the use of a low cost robot, personal digital assistant (PDA) handheld computer, and custom PDA software. In this course, the concepts and theories discussed in lecture are reinforced and extended in a corresponding laboratory through the use of wireless robots and PDAs. Fuzzy logic libraries and software modules for laptops and desktop computers are readily available, however, when it comes to handheld computers no such libraries exist. Students are able to spend more time experimenting with different fuzzy logic controllers when a custom fuzzy logic library and PDA graphical user interface are utilized. In this paper we introduce and discuss a unique low cost wireless robot, a custom fuzzy logic library, a custom fuzzy logic GUI for the PDA, and the implementation results for the fuzzy logic section in a newly created bioengineering course. Diagnostic and summative assessment in the form of a pre-test and post-test was administered for each section of the course, however, only the results for the fuzzy logic section will be provided.

By employing a recently obtained error bound for differentiable convex inequalities, it is shown that, under appropriate constraint qualifications, a minimum solution of an exact penalty function for a single value of the penalty parameter which exceeds a certain threshold, is also a solution of the convex program associated with the penalty function. No a priori assumption is made regarding the solvability of the convex program. If such a solvability assumption is made, then we show that a threshold value of the penalty parameter can be used which is smaller than both the above-mentioned value and that of Zangwill. These various threshold values of the penalty parameter also apply to the well-known big-M method of linear programming.