Mathematical Programming最新文献

Mixed-integer nonlinear optimization formulations of the disjunction between the origin and a polytope via a binary indicator variable is broadly used in nonlinear combinatorial optimization for modeling a fixed cost associated with carrying out a group of activities and a convex cost function associated with the levels of the activities. The perspective relaxation of such models is often used to solve to global optimality in a branch-and-bound context, but it typically requires suitable conic solvers and is not compatible with general-purpose NLP software in the presence of other classes of constraints. This motivates the investigation of when simpler but weaker relaxations may be adequate. Comparing the volume (i.e., Lebesgue measure) of the relaxations as a measure of tightness, we lift some of the results related to the simplex case to the box case. In order to compare the volumes of different relaxations in the box case, it is necessary to find an appropriate concave upper bound that preserves the convexity and is minimal, which is more difficult than in the simplex case. To address the challenge beyond the simplex case, the triangulation approach is used.

Several classical adaptive optimization algorithms, such as line search and trust-region methods, have been recently extended to stochastic settings where function values, gradients, and Hessians in some cases, are estimated via stochastic oracles. Unlike the majority of stochastic methods, these methods do not use a pre-specified sequence of step size parameters, but adapt the step size parameter according to the estimated progress of the algorithm and use it to dictate the accuracy required from the stochastic oracles. The requirements on the stochastic oracles are, thus, also adaptive and the oracle costs can vary from iteration to iteration. The step size parameters in these methods can increase and decrease based on the perceived progress, but unlike the deterministic case they are not bounded away from zero due to possible oracle failures, and bounds on the step size parameter have not been previously derived. This creates obstacles in the total complexity analysis of such methods, because the oracle costs are typically decreasing in the step size parameter, and could be arbitrarily large as the step size parameter goes to 0. Thus, until now only the total iteration complexity of these methods has been analyzed. In this paper, we derive a lower bound on the step size parameter that holds with high probability for a large class of adaptive stochastic methods. We then use this lower bound to derive a framework for analyzing the expected and high probability total oracle complexity of any method in this class. Finally, we apply this framework to analyze the total sample complexity of two particular algorithms, STORM (Blanchet et al. in INFORMS J Optim 1(2):92–119, 2019) and SASS (Jin et al. in High probability complexity bounds for adaptive step search based on stochastic oracles, 2021. https://doi.org/10.48550/ARXIV.2106.06454), in the expected risk minimization problem.

Approximate integer programming is the following: For a given convex body (K subseteq {mathbb {R}}^n), either determine whether (K cap {mathbb {Z}}^n) is empty, or find an integer point in the convex body (2cdot (K - c) +c) which is K, scaled by 2 from its center of gravity c. Approximate integer programming can be solved in time (2^{O(n)}) while the fastest known methods for exact integer programming run in time (2^{O(n)} cdot n^n). So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point (x^* in (K cap {mathbb {Z}}^n)) can be found in time (2^{O(n)}), provided that the remainders of each component (x_i^* mod ell ) for some arbitrarily fixed (ell ge 5(n+1)) of (x^*) are given. The algorithm is based on a cutting-plane technique, iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a (2^{O(n)}n^n) algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a new asymmetric approximate Carathéodory theorem that might be of interest on its own. Our second method concerns integer programming problems in equation-standard form (Ax = b, 0 le x le u, , x in {mathbb {Z}}^n). Such a problem can be reduced to the solution of (prod _i O(log u_i +1)) approximate integer programming problems. This implies, for example that knapsack or subset-sum problems with polynomial variable range (0 le x_i le p(n)) can be solved in time ((log n)^{O(n)}). For these problems, the best running time so far was (n^n cdot 2^{O(n)}).

We consider the minimum-norm-point (MNP) problem over polyhedra, a well-studied problem that encompasses linear programming. We present a general algorithmic framework that combines two fundamental approaches for this problem: active set methods and first order methods. Our algorithm performs first order update steps, followed by iterations that aim to ‘stabilize’ the current iterate with additional projections, i.e., find a locally optimal solution whilst keeping the current tight inequalities. Such steps have been previously used in active set methods for the nonnegative least squares (NNLS) problem. We bound on the number of iterations polynomially in the dimension and in the associated circuit imbalance measure. In particular, the algorithm is strongly polynomial for network flow instances. Classical NNLS algorithms such as the Lawson–Hanson algorithm are special instantiations of our framework; as a consequence, we obtain convergence bounds for these algorithms. Our preliminary computational experiments show promising practical performance.

We extend the convergence analysis of the Scholtes-type regularization method for cardinality-constrained optimization problems. Its behavior is clarified in the vicinity of saddle points, and not just of minimizers as it has been done in the literature before. This becomes possible by using as an intermediate step the recently introduced regularized continuous reformulation of a cardinality-constrained optimization problem. We show that the Scholtes-type regularization method is well-defined locally around a nondegenerate T-stationary point of this regularized continuous reformulation. Moreover, the nondegenerate Karush–Kuhn–Tucker points of the corresponding Scholtes-type regularization converge to a T-stationary point having the same index, i.e. its topological type persists. As consequence, we conclude that the global structure of the Scholtes-type regularization essentially coincides with that of CCOP.

A branch-and-bound (BB) tree certifies a dual bound on the value of an integer program. In this work, we introduce the tree compression problem (TCP): Given a BB tree T that certifies a dual bound, can we obtain a smaller tree with the same (or stronger) bound by either (1) applying a different disjunction at some node in T or (2) removing leaves from T? We believe such post-hoc analysis of BB trees may assist in identifying helpful general disjunctions in BB algorithms. We initiate our study by considering computational complexity and limitations of TCP. We then conduct experiments to evaluate the compressibility of realistic branch-and-bound trees generated by commonly-used branching strategies, using both an exact and a heuristic compression algorithm.

We consider the global minimization of smooth functions based solely on function evaluations. Algorithms that achieve the optimal number of function evaluations for a given precision level typically rely on explicitly constructing an approximation of the function which is then minimized with algorithms that have exponential running-time complexity. In this paper, we consider an approach that jointly models the function to approximate and finds a global minimum. This is done by using infinite sums of square smooth functions and has strong links with polynomial sum-of-squares hierarchies. Leveraging recent representation properties of reproducing kernel Hilbert spaces, the infinite-dimensional optimization problem can be solved by subsampling in time polynomial in the number of function evaluations, and with theoretical guarantees on the obtained minimum. Given n samples, the computational cost is (O(n^{3.5})) in time, (O(n^2)) in space, and we achieve a convergence rate to the global optimum that is (O(n^{-m/d + 1/2 + 3/d})) where m is the degree of differentiability of the function and d the number of dimensions. The rate is nearly optimal in the case of Sobolev functions and more generally makes the proposed method particularly suitable for functions with many derivatives. Indeed, when m is in the order of d, the convergence rate to the global optimum does not suffer from the curse of dimensionality, which affects only the worst-case constants (that we track explicitly through the paper).

Abstract

Risk measures are commonly used to capture the risk preferences of decision-makers (DMs). The decisions of DMs can be nudged or manipulated when their risk preferences are influenced by factors such as the availability of information about the uncertainties. This work proposes a Stackelberg risk preference design (STRIPE) problem to capture a designer’s incentive to influence DMs’ risk preferences. STRIPE consists of two levels. In the lower level, individual DMs in a population, known as the followers, respond to uncertainties according to their risk preference types. In the upper level, the leader influences the distribution of the types to induce targeted decisions and steers the follower’s preferences to it. Our analysis centers around the solution concept of approximate Stackelberg equilibrium that yields suboptimal behaviors of the players. We show the existence of the approximate Stackelberg equilibrium. The primitive risk perception gap, defined as the Wasserstein distance between the original and the target type distributions, is important in estimating the optimal design cost. We connect the leader’s optimality compromise on the cost with her ambiguity tolerance on the follower’s approximate solutions leveraging Lipschitzian properties of the lower level solution mapping. To obtain the Stackelberg equilibrium, we reformulate STRIPE into a single-level optimization problem using the spectral representations of law-invariant coherent risk measures. We create a data-driven approach for computation and study its performance guarantees. We apply STRIPE to contract design problems under approximate incentive compatibility. Moreover, we connect STRIPE with meta-learning problems and derive adaptation performance estimates of the meta-parameters.

Mental illnesses are the leading cause of disease burden among children and young people (CYP) globally. Low- and middle-income countries (LMIC) are disproportionately affected. Enhancing mental health literacy (MHL) is one way to combat low levels of help-seeking and effective treatment receipt. We aimed to synthesis evidence about knowledge, beliefs and attitudes of CYP in LMICs about mental illnesses, their treatments and outcomes, evaluating factors that can enhance or impede help-seeking to inform context-specific and developmentally appropriate understandings of MHL. Eight bibliographic databases were searched from inception to July 2020: PsycInfo, EMBASE, Medline (OVID), Scopus, ASSIA (ProQuest), SSCI, SCI (Web of Science) CINAHL PLUS, Social Sciences full text (EBSCO). 58 papers (41 quantitative, 13 qualitative, 4 mixed methods) representing 52 separate studies comprising 36,429 participants with a mean age of 15.3 [10.4-17.4], were appraised and synthesized using narrative synthesis methods. Low levels of recognition and knowledge about mental health problems and illnesses, pervasive levels of stigma and low confidence in professional healthcare services, even when considered a valid treatment option were dominant themes. CYP cited the value of traditional healers and social networks for seeking help. Several important areas were under-researched including the link between specific stigma types and active help-seeking and research is needed to understand more fully the interplay between knowledge, beliefs and attitudes across varied cultural settings. Greater exploration of social networks and the value of collaboration with traditional healers is consistent with promising, yet understudied, areas of community-based MHL interventions combining education and social contact.

Abstract

This work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies based on the Schmüdgen and Putinar Positivstellensätze enjoy a polynomial rate of convergence that depends on the size of the largest clique in the sparsity graph but not on the ambient dimension. Interestingly, the sparse bounds outperform the best currently available bounds for the dense hierarchy when the maximum clique size is sufficiently small compared to the ambient dimension and the performance is measured by the running time of an interior point method required to obtain a bound on the global minimum of a given accuracy.