Mathematical Programming最新文献

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.

This paper introduces a new storage-optimal first-order method, CertSDP, for solving a special class of semidefinite programs (SDPs) to high accuracy. The class of SDPs that we consider, the exact QMP-like SDPs, is characterized by low-rank solutions, a priori knowledge of the restriction of the SDP solution to a small subspace, and standard regularity assumptions such as strict complementarity. Crucially, we show how to use a certificate of strict complementarity to construct a low-dimensional strongly convex minimax problem whose optimizer coincides with a factorization of the SDP optimizer. From an algorithmic standpoint, we show how to construct the necessary certificate and how to solve the minimax problem efficiently. Our algorithms for strongly convex minimax problems with inexact prox maps may be of independent interest. We accompany our theoretical results with preliminary numerical experiments suggesting that CertSDP significantly outperforms current state-of-the-art methods on large sparse exact QMP-like SDPs.

Abstract

We consider min–max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables. When the variables belong to simple sets (e.g., a hypercube, the Euclidean hypersphere, or a ball), we derive a sum-of-squares formulation based on a primal-dual approach. In the simplest setting, we provide a convergence proof when the degree of the relaxation tends to infinity and observe empirically that it can be finitely convergent in several situations. Moreover, our formulation leads to an interesting link with feasibility certificates for polynomial inequalities based on Putinar’s Positivstellensatz.

Globally optimizing a nonconvex quadratic over the intersection of m balls in (mathbb {R}^n) is known to be polynomial-time solvable for fixed m. Moreover, when (m=1), the standard semidefinite relaxation is exact. When (m=2), it has been shown recently that an exact relaxation can be constructed using a disjunctive semidefinite formulation based essentially on two copies of the (m=1) case. However, there is no known explicit, tractable, exact convex representation for (m ge 3). In this paper, we construct a new, polynomially sized semidefinite relaxation for all m, which does not employ a disjunctive approach. We show that our relaxation is exact for (m=2). Then, for (m ge 3), we demonstrate empirically that it is fast and strong compared to existing relaxations. The key idea of the relaxation is a simple lifting of the original problem into dimension (n, +, 1). Extending this construction: (i) we show that nonconvex quadratic programming over (Vert xVert le min { 1, g + h^T x }) has an exact semidefinite representation; and (ii) we construct a new relaxation for quadratic programming over the intersection of two ellipsoids, which globally solves all instances of a benchmark collection from the literature.

This paper settles an open and challenging question pertaining to the design of simple and optimal high-order methods for solving smooth and monotone variational inequalities (VIs). A VI involves finding (x^star in {mathcal {X}}) such that (langle F(x), x - x^star rangle ge 0) for all (x in {mathcal {X}}). We consider the setting in which (F: {mathbb {R}}^d rightarrow {mathbb {R}}^d) is smooth with up to ((p-1)^{text {th}})-order derivatives. For (p = 2), the cubic regularization of Newton’s method has been extended to VIs with a global rate of (O(epsilon ^{-1})) (Nesterov in Cubic regularization of Newton’s method for convex problems with constraints, Tech. rep., Université catholique de Louvain, Center for Operations Research and Econometrics (CORE), 2006). An improved rate of (O(epsilon ^{-2/3}log log (1/epsilon ))) can be obtained via an alternative second-order method, but this method requires a nontrivial line-search procedure as an inner loop. Similarly, the existing high-order methods based on line-search procedures have been shown to achieve a rate of (O(epsilon ^{-2/(p+1)}log log (1/epsilon ))) (Bullins and Lai in SIAM J Optim 32(3):2208–2229, 2022; Jiang and Mokhtari in Generalized optimistic methods for convex–concave saddle point problems, 2022; Lin and Jordan in Math Oper Res 48(4):2353–2382, 2023). As emphasized by Nesterov (Lectures on convex optimization, vol 137, Springer, Berlin, 2018), however, such procedures do not necessarily imply the practical applicability in large-scale applications, and it is desirable to complement these results with a simple high-order VI method that retains the optimality of the more complex methods. We propose a (p^{text {th}})-order method that does not require any line search procedure and provably converges to a weak solution at a rate of (O(epsilon ^{-2/(p+1)})). We prove that our (p^{text {th}})-order method is optimal in the monotone setting by establishing a lower bound of (Omega (epsilon ^{-2/(p+1)})) under a generalized linear span assumption. A restarted version of our (p^{text {th}})-order method attains a linear rate for smooth and (p^{text {th}})-order uniformly monotone VIs and another restarted version of our (p^{text {th}})-order method attains a local superlinear rate for smooth and strongly monotone VIs. Further, the similar (p^{text {th}})-order method achieves a global rate of (O(epsilon ^{-2/p})) for solving smooth and nonmonotone VIs satisfying the Minty condition. Two restarted versions attain a global linear rate under additional (p^{text {th}})-order uniform Minty condition and a local superlinear rate under additional strong Minty condition.