Mathematical Programming最新文献

A set function can be extended to the unit cube in various ways; the correlation gap measures the ratio between two natural extensions. This quantity has been identified as the performance guarantee in a range of approximation algorithms and mechanism design settings. It is known that the correlation gap of a monotone submodular function is at least (1-1/e), and this is tight for simple matroid rank functions. We initiate a fine-grained study of the correlation gap of matroid rank functions. In particular, we present an improved lower bound on the correlation gap as parametrized by the rank and girth of the matroid. We also show that for any matroid, the correlation gap of its weighted rank function is minimized under uniform weights. Such improved lower bounds have direct applications for submodular maximization under matroid constraints, mechanism design, and contention resolution schemes.

The configuration balancing problem with stochastic requests generalizes well-studied resource allocation problems such as load balancing and virtual circuit routing. There are given m resources and n requests; each request has multiple possible configurations, each of which increases the load of each resource by some amount. The goal is to select one configuration for each request to minimize the makespan: the load of the most-loaded resource. In the stochastic setting, the amount by which a configuration increases the resource load is uncertain until the configuration is chosen, but we are given a probability distribution. We develop both offline and online algorithms for configuration balancing with stochastic requests. When the requests are known offline, we give a non-adaptive policy for configuration balancing with stochastic requests that (O(frac{log m}{log log m}))-approximates the optimal adaptive policy, which matches a known lower bound for the special case of load balancing on identical machines. When requests arrive online in a list, we give a non-adaptive policy that is (O(log m)) competitive. Again, this result is asymptotically tight due to information-theoretic lower bounds for special cases (e.g., for load balancing on unrelated machines). Finally, we show how to leverage adaptivity in the special case of load balancing on related machines to obtain a constant-factor approximation offline and an (O(log log m))-approximation online. A crucial technical ingredient in all of our results is a new structural characterization of the optimal adaptive policy that allows us to limit the correlations between its decisions.

In this paper, we focus on a class of convexly constrained nonsmooth convex–concave saddle point problems with cardinality penalties. Although such nonsmooth nonconvex–nonconcave and discontinuous min–max problems may not have a saddle point, we show that they have a local saddle point and a global minimax point, and some local saddle points have the lower bound properties. We define a class of strong local saddle points based on the lower bound properties for stability of variable selection. Moreover, we give a framework to construct continuous relaxations of the discontinuous min–max problems based on convolution, such that they have the same saddle points with the original problem. We also establish the relations between the continuous relaxation problems and the original problems regarding local saddle points, global minimax points, local minimax points and stationary points. Finally, we illustrate our results with distributionally robust sparse convex regression, sparse robust bond portfolio construction and sparse convex–concave logistic regression saddle point problems.

Strongly motivated from applications in various fields including machine learning, the methodology of sparse optimization has been developed intensively so far. Especially, the advancement of algorithms for solving problems with nonsmooth regularizers has been remarkable. However, those algorithms suppose that weight parameters of regularizers, called hyperparameters hereafter, are pre-fixed, but it is a crucial matter how the best hyperparameter should be selected. In this paper, we focus on the hyperparameter selection of regularizers related to the (ell _p) function with (0<ple 1) and apply a bilevel programming strategy, wherein we need to solve a bilevel problem, whose lower-level problem is nonsmooth, possibly nonconvex and non-Lipschitz. Recently, for solving a bilevel problem for hyperparameter selection of the pure (ell _p (0<p le 1)) regularizer Okuno et al. discovered new necessary optimality conditions, called SB(scaled bilevel)-KKT conditions, and further proposed a smoothing-type algorithm using a specific smoothing function. However, this optimality measure is loose in the sense that there could be many points that satisfy the SB-KKT conditions. In this work, we propose new bilevel KKT conditions, which are new necessary optimality conditions tighter than the ones proposed by Okuno et al. Moreover, we propose a unified smoothing approach using smoothing functions that belong to the Chen-Mangasarian class, and then prove that generated iteration points accumulate at bilevel KKT points under milder constraint qualifications. Another contribution is that our approach and analysis are applicable to a wider class of regularizers. Numerical comparisons demonstrate which smoothing functions work well for hyperparameter optimization via bilevel optimization approach.

We consider the problem of designing a linear program that has diverse solutions as the right-hand side varies. This problem arises in video game settings where designers aim to have players use different “weapons” or “tactics” as they progress. We model this design question as a choice over the constraint matrix A and cost vector c to maximize the number of possible supports of unique optimal solutions (what we call “loadouts”) of Linear Programs (max {c^top x mid Ax le b, x ge 0}) with nonnegative data considered over all resource vectors b. We provide an upper bound on the optimal number of loadouts and provide a family of constructions that have an asymptotically optimal number of loadouts. The upper bound is based on a connection between our problem and the study of triangulations of point sets arising from polyhedral combinatorics, and specifically the combinatorics of the cyclic polytope. Our asymptotically optimal construction also draws inspiration from the properties of the cyclic polytope.

We present a novel response surface method for global optimisation of an expensive and noisy (black-box) objective function, where error bounds on the deviation of the observed noisy function values from their true counterparts are available. The method is based on Gutmann’s well-established RBF method for minimising an expensive and deterministic objective function, which has become popular both from a theoretical and practical perspective. To construct suitable radial basis function approximants to the objective function and to determine new sample points for successive evaluation of the expensive noisy objective, the method uses a regularised least-squares criterion. In particular, new points are defined by means of a target value, analogous to the original RBF method. We provide essential convergence results, and provide a numerical illustration of the method by means of a simple test problem.

Optimization of distortion riskmetrics with distributional uncertainty has wide applications in finance and operations research. Distortion riskmetrics include many commonly applied risk measures and deviation measures, which are not necessarily monotone or convex. One of our central findings is a unifying result that allows to convert an optimization of a non-convex distortion riskmetric with distributional uncertainty to a convex one induced from the concave envelope of the distortion function, leading to practical tractability. A sufficient condition to the unifying equivalence result is the novel notion of closedness under concentration, a variation of which is also shown to be necessary for the equivalence. Our results include many special cases that are well studied in the optimization literature, including but not limited to optimizing probabilities, Value-at-Risk, Expected Shortfall, Yaari’s dual utility, and differences between distortion risk measures, under various forms of distributional uncertainty. We illustrate our theoretical results via applications to portfolio optimization, optimization under moment constraints, and preference robust optimization.

We present a Markov-chain analysis of blockwise-stochastic algorithms for solving partially block-separable optimization problems. Our main contributions to the extensive literature on these methods are statements about the Markov operators and distributions behind the iterates of stochastic algorithms, and in particular the regularity of Markov operators and rates of convergence of the distributions of the corresponding Markov chains. This provides a detailed characterization of the moments of the sequences beyond just the expected behavior. This also serves as a case study of how randomization restores favorable properties to algorithms that iterations of only partial information destroys. We demonstrate this on stochastic blockwise implementations of the forward–backward and Douglas–Rachford algorithms for nonconvex (and, as a special case, convex), nonsmooth optimization.

Supervalid inequalities are a specific type of constraints often used within the branch-and-cut framework to strengthen the linear relaxation of mixed-integer programs. These inequalities share the particular characteristic of potentially removing feasible integer solutions as long as they are already dominated by an incumbent solution. This paper focuses on supervalid inequalities for solving binary interdiction games. Specifically, we provide a general characterization of inequalities that are derived from bipartitions of the leader’s strategy set and develop an algorithmic approach to use them. This includes the design of two verification subroutines that we apply for separation purposes. We provide three general examples in which we apply our results to solve binary interdiction games targeting shortest paths, spanning trees, and vertex covers. Finally, we prove that the separation procedure is efficient for the class of interdiction games defined on greedoids—a type of set system that generalizes many others such as matroids and antimatroids.

With the goal of obtaining strong relaxations for binary polynomial optimization problems, we introduce the pseudo-Boolean polytope defined as the set of binary points (z in {0,1}^{V cup S}) satisfying a collection of equalities of the form (z_s = prod _{v in s} sigma _s(z_v)), for all (s in S), where (sigma _s(z_v) in {z_v, 1-z_v}), and where S is a multiset of subsets of V. By representing the pseudo-Boolean polytope via a signed hypergraph, we obtain sufficient conditions under which this polytope has a polynomial-size extended formulation. Our new framework unifies and extends all prior results on the existence of polynomial-size extended formulations for the convex hull of the feasible region of binary polynomial optimization problems of degree at least three.