Mathematical Programming最新文献

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.

Abstract

We consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido–Isoda function. To any given instance of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.

Abstract

In this paper we study the well-known Chvátal–Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.

Given a fixed finite metric space ((V,mu )), the minimum 0-extension problem, denoted as (mathtt{0hbox {-}Ext}[{mu }]), is equivalent to the following optimization problem: minimize function of the form (min nolimits _{xin V^n} sum _i f_i(x_i) + sum _{ij} c_{ij}hspace{0.5pt}mu (x_i,x_j)) where (f_i:Vrightarrow mathbb {R}) are functions given by (f_i(x_i)=sum _{vin V} c_{vi}hspace{0.5pt}mu (x_i,v)) and (c_{ij},c_{vi}) are given nonnegative costs. The computational complexity of (mathtt{0hbox {-}Ext}[{mu }]) has been recently established by Karzanov and by Hirai: if metric (mu ) is orientable modular then (mathtt{0hbox {-}Ext}[{mu }]) can be solved in polynomial time, otherwise (mathtt{0hbox {-}Ext}[{mu }]) is NP-hard. To prove the tractability part, Hirai developed a theory of discrete convex functions on orientable modular graphs generalizing several known classes of functions in discrete convex analysis, such as (L^natural )-convex functions. We consider a more general version of the problem in which unary functions (f_i(x_i)) can additionally have terms of the form (c_{uv;i}hspace{0.5pt}mu (x_i,{u,v})) for ({u,!hspace{0.5pt}hspace{0.5pt}v}in F), where set (Fsubseteq left( {begin{array}{c}V 2end{array}}right) ) is fixed. We extend the complexity classification above by providing an explicit condition on ((mu ,F)) for the problem to be tractable. In order to prove the tractability part, we generalize Hirai’s theory and define a larger class of discrete convex functions. It covers, in particular, another well-known class of functions, namely submodular functions on an integer lattice. Finally, we improve the complexity of Hirai’s algorithm for solving (mathtt{0hbox {-}Ext}[{mu }]) on orientable modular graphs.

We present a new feasible proximal gradient method for constrained optimization where both the objective and constraint functions are given by summation of a smooth, possibly nonconvex function and a convex simple function. The algorithm converts the original problem into a sequence of convex subproblems. Formulating those subproblems requires the evaluation of at most one gradient-value of the original objective and constraint functions. Either exact or approximate subproblems solutions can be computed efficiently in many cases. An important feature of the algorithm is the constraint level parameter. By carefully increasing this level for each subproblem, we provide a simple solution to overcome the challenge of bounding the Lagrangian multipliers and show that the algorithm follows a strictly feasible solution path till convergence to the stationary point. We develop a simple, proximal gradient descent type analysis, showing that the complexity bound of this new algorithm is comparable to gradient descent for the unconstrained setting which is new in the literature. Exploiting this new design and analysis technique, we extend our algorithms to some more challenging constrained optimization problems where (1) the objective is a stochastic or finite-sum function, and (2) structured nonsmooth functions replace smooth components of both objective and constraint functions. Complexity results for these problems also seem to be new in the literature. Finally, our method can also be applied to convex function constrained problems where we show complexities similar to the proximal gradient method.

We study linear programming relaxations of nonconvex quadratic programs given by the reformulation–linearization technique (RLT), referred to as RLT relaxations. We investigate the relations between the polyhedral properties of the feasible regions of a quadratic program and its RLT relaxation. We establish various connections between recession directions, boundedness, and vertices of the two feasible regions. Using these properties, we present a complete description of the set of instances that admit an exact RLT relaxation. We then give a thorough discussion of how our results can be converted into simple algorithmic procedures to construct instances of quadratic programs with exact, inexact, or unbounded RLT relaxations.

This paper aims to find an approximate true sparse solution of an underdetermined linear system. For this purpose, we propose two types of iterative thresholding algorithms with the continuation technique and the truncation technique respectively. We introduce a notion of limited shrinkage thresholding operator and apply it, together with the restricted isometry property, to show that the proposed algorithms converge to an approximate true sparse solution within a tolerance relevant to the noise level and the limited shrinkage magnitude. Applying the obtained results to nonconvex regularization problems with SCAD, MCP and (ell _p) penalty ((0le p le 1)) and utilizing the recovery bound theory, we establish the convergence of their proximal gradient algorithms to an approximate global solution of nonconvex regularization problems. The established results include the existing convergence theory for (ell _1) or (ell _0) regularization problems for finding a true sparse solution as special cases. Preliminary numerical results show that our proposed algorithms can find approximate true sparse solutions that are much better than stationary solutions that are found by using the standard proximal gradient algorithm.