An explicit step length for solving an optimization problem

In this paper, we introduce a transformation that converts a class of linear and nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems. For those problems of interest, the transformation replaces matrix-valued constraints by vector-valued ones, hence reducing the number of constraints by an order of magnitude. The class of transformable problems includes instances of SDP relaxations of combinatorial optimization problems with binary variables as well as other important SDP problems. We also derive gradient formulas for the objective function of the resulting nonlinear optimization problem and show that both function and gradient evaluations have affordable complexities that effectively exploit the sparsity of the problem data. This transformation, together with the efficient gradient formulas, enables the solution of very large-scale SDP problems by gradient-based nonlinear optimization techniques. In particular, we propose a first-order log-barrier method designed for solving a class of large-scale linear SDP problems. This algorithm operates entirely within the space of the transformed problem while still maintaining close ties with both the primal and the dual of the original SDP problem. Global convergence of the algorithm is established under mild and reasonable assumptions.

The semi definite programming (SDP) problem is one of the central problems in mathematical optimization. The primal-dual interior-point method (PDIPM) is one of the most powerful algorithms for solving SDP problems, and many research groups have employed it for developing software packages. However, two well-known major bottlenecks, i.e., the generation of the Schur complement matrix (SCM) and its Cholesky factorization, exist in the algorithmic framework of the PDIPM. We have developed a new version of the semi definite programming algorithm parallel version (SDPARA), which is a parallel implementation on multiple CPUs and GPUs for solving extremely large-scale SDP problems with over a million constraints. SDPARA can automatically extract the unique characteristics from an SDP problem and identify the bottleneck. When the generation of the SCM becomes a bottleneck, SDPARA can attain high scalability using a large quantity of CPU cores and some processor affinity and memory interleaving techniques. SDPARA can also perform parallel Cholesky factorization using thousands of GPUs and techniques for overlapping computation and communication if an SDP problem has over two million constraints and Cholesky factorization constitutes a bottleneck. We demonstrate that SDPARA is a high-performance general solver for SDPs in various application fields through numerical experiments conducted on the TSUBAME 2.5 supercomputer, and we solved the largest SDP problem (which has over 2.33 million constraints), thereby creating a new world record. Our implementation also achieved 1.713 PFlops in double precision for large-scale Cholesky factorization using 2,720 CPUs and 4,080 GPUs.

Second-order cone programming (SOCP) problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of second-order (Lorentz) cones. Linear programs, convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as SOCP problems, as can many other problems that do not fall into these three categories. These latter problems model applications from a broad range of fields from engineering, control and finance to robust optimization and combinatorial optimization. On the other hand semidefinite programming (SDP)—that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices—includes SOCP as a special case. Therefore, SOCP falls between linear (LP) and quadratic (QP) programming and SDP. Like LP, QP and SDP problems, SOCP problems can be solved in polynomial time by interior point methods. The computational effort per iteration required by these methods to solve SOCP problems is greater than that required to solve LP and QP problems but less than that required to solve SDP’s of similar size and structure. Because the set of feasible solutions for an SOCP problem is not polyhedral as it is for LP and QP problems, it is not readily apparent how to develop a simplex or simplex-like method for SOCP. While SOCP problems can be solved as SDP problems, doing so is not advisable both on numerical grounds and computational complexity concerns. For instance, many of the problems presented in the survey paper of Vandenberghe and Boyd [VB96] as examples of SDPs can in fact be formulated as SOCPs and should be solved as such. In §2, 3 below we give SOCP formulations for four of these examples: the convex quadratically constrained quadratic programming (QCQP) problem, problems involving fractional quadratic functions ∗RUTCOR, Rutgers University, e-mail:alizadeh@rutcor.rutgers.edu. Research supported in part by the U.S. National Science Foundation grant CCR-9901991 †IEOR, Columbia University, e-mail: gold@ieor.columbia.edu. Research supported in part by the Department of Energy grant DE-FG02-92ER25126, National Science Foundation grants DMS-94-14438, CDA-97-26385 and DMS-01-04282.

In this talk, we present our ongoing research project. The objective of this project is to develop advanced computing and optimization infrastructures for extremely large-scale graphs on post peta-scale supercomputers. We explain our challenge to Graph 500 and Green Graph 500 benchmarks that are designed to measure the performance of a computer system for applications that require irregular memory and network access patterns. The 1st Graph500 list was released in November 2010. The Graph500 benchmark measures the performance of any supercomputer performing a BFS (Breadth-First Search) in terms of traversed edges per second (TEPS). In 2014 and 2015, our project team was a winner of the 8th, 10th, and 11th Graph500 and the 3rd to 6th Green Graph500 benchmarks, respectively. We also present our parallel implementation for large-scale SDP (SemiDefinite Programming) problem. The semidefinite programming (SDP) problem is a predominant problem in mathematical optimization. The primal-dual interior-point method (PDIPM) is one of the most powerful algorithms for solving SDP problems, and many research groups have employed it for developing software packages. We solved the largest SDP problem (which has over 2.33 million constraints), thereby creating a new world record. Our implementation also achieved 1.774 PFlops in double precision for large-scale Cholesky factorization using 2,720 CPUs and 4,080 GPUs on the TSUBAME 2.5 supercomputer.

The semidefinite programming (SDP) problem is one of the central problems in mathematical optimization. The primal-dual interior-point method (PDIPM) is one of the most powerful algorithms for solving SDP problems, and many research groups have employed it for developing software packages. However, two well-known major bottlenecks, i.e., the generation of the Schur complement matrix (SCM) and its Cholesky factorization, exist in the algorithmic framework of the PDIPM. We have developed a new version of the semidefinite programming algorithm parallel version (SDPARA), which is a parallel implementation on multiple CPUs and GPUs for solving extremely large-scale SDP problems with over a million constraints. SDPARA can automatically extract the unique characteristics from an SDP problem and identify the bottleneck. When the generation of the SCM becomes a bottleneck, SDPARA can attain high scalability using a large quantity of CPU cores and some processor affinity and memory interleaving techniques. SDPARA can also perform parallel Cholesky factorization using thousands of GPUs and techniques for overlapping computation and communication if an SDP problem has over 2 million constraints and Cholesky factorization constitutes a bottleneck. We demonstrate that SDPARA is a high-performance general solver for SDPs in various application fields through numerical experiments conducted on the TSUBAME 2.5 supercomputer, and we solved the largest SDP problem (which has over 2.33 million constraints), thereby creating a new world record. Our implementation also achieved 1.713 PFlops in double precision for large-scale Cholesky factorization using 2,720 CPUs and 4,080 GPUs.

Solving optimization problems with sparse or low-rank optimal solutions has been an important topic since the recent emergence of compressed sensing and its matrix extensions such as the matrix rank minimization and robust principal component analysis problems. Compressed sensing enables one to recover a signal or image with fewer observations than the “length” of the signal or image, and thus provides potential breakthroughs in applications where data acquisition is costly. However, the potential impact of compressed sensing cannot be realized without efficient optimization algorithms that can handle extremely large-scale and dense data from real applications. Although the convex relaxations of these problems can be reformulated as either linear programming, second-order cone programming or semidefinite programming problems, the standard methods for solving these relaxations are not applicable because the problems are usually of huge size and contain dense data. In this dissertation, we give efficient algorithms for solving these “sparse” optimization problems and analyze the convergence and iteration complexity properties of these algorithms. Chapter 2 presents algorithms for solving the linearly constrained matrix rank minimization problem. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast and solved as a semidefinite programming problem, such an approach is computationally expensive when the matrices are large. In Chapter 2, we propose fixed-point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10−5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. In Chapter 3, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving linearly constrained matrix rank minimization problems are reported. Chapters 4 and 5 considers alternating direction type methods for solving composite convex optimization problems. We present in Chapter 4 alternating linearization algorithms that are based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most O(1/e) iterations to obtain an e-optimal solution, while our accelerated (i.e., fast) versions require at most O(1/ 3 ) iterations, with little change in the computational effort required at each iteration. For more general problem, i.e., minimizing the sum of K convex functions, we propose multiple-splitting algorithms for solving them. We propose both basic and accelerated algorithms with O(1/e) and O(1/ 3 ) iteration complexity bounds for obtaining an e-optimal solution. To the best of our knowledge, the complexity results presented in these two chapters are the first ones of this type that have been given for splitting and alternating direction type methods. Numerical results on various applications in sparse and low-rank optimization, including compressed sensing, matrix completion, image deblurring, robust principal component analysis, are reported to demonstrate the efficiency of our methods.

Modal dynamic residual-based model updating through regularized semidefinite programming with facial reduction

This paper presents a study of regularity of Semidefinite Programming (SDP) problems. Current methods for SDP rely on assumptions of regularity such as constraint qualifications (CQ) and well-posedness. In the absence of regularity, the characterization of optimality may fail and the convergence of algorithms is not guaranteed. Therefore, it is important to have procedures that verify the regularity of a given problem before applying any (standard) SDP solver. We suggest a simple numerical procedure to test within a desired accuracy if a given SDP problem is regular in terms of the fulfilment of the Slater CQ. Our procedure is based on the recently proposed DIIS algorithm that determines the immobile index subspace for SDP. We use this algorithm in a framework of an interactive decision support system. Numerical results using SDP problems from the literature and instances from the SDPLIB suite are presented, and a comparative analysis with other results on regularity available in the literature is made.

Exact SDP relaxations for classes of nonlinear semidefinite programming problems

In this paper, we consider a least square semidefinite programming problem under ellipsoidal data uncertainty. We show that the robustification of this uncertain problem can be reformulated as a semidefinite linear programming problem with an additional second-order cone constraint. We then provide an explicit quantitative sensitivity analysis on how the solution under the robustification depends on the size/shape of the ellipsoidal data uncertainty set. Next, we prove that, under suitable constraint qualifications, the reformulation has zero duality gap with its dual problem, even when the primal problem itself is infeasible. The dual problem is equivalent to minimizing a smooth objective function over the Cartesian product of second-order cones and the Euclidean space, which is easy to project onto. Thus, we propose a simple variant of the spectral projected gradient method (Birgin et al. in SIAM J. Optim. 10:1196---1211, 2000) to solve the dual problem. While it is well-known that any accumulation point of the sequence generated from the algorithm is a dual optimal solution, we show in addition that the dual objective value along the sequence generated converges to a finite value if and only if the primal problem is feasible, again under suitable constraint qualifications. This latter fact leads to a simple certificate for primal infeasibility in situations when the primal feasible set lies in a known compact set. As an application, we consider robust correlation stress testing where data uncertainty arises due to untimely recording of portfolio holdings. In our computational experiments on this particular application, our algorithm performs reasonably well on medium-sized problems for real data when finding the optimal solution (if exists) or identifying primal infeasibility, and usually outperforms the standard interior-point solver SDPT3 in terms of CPU time.

Random projection, a dimensionality reduction technique, has been found useful in recent years for reducing the size of optimization problems. In this paper, we explore the use of sparse sub-gaussian random projections to approximate semidefinite programming (SDP) problems by reducing the size of matrix variables, thereby solving the original problem with much less computational effort. We provide some theoretical bounds on the quality of the projection in terms of feasibility and optimality that explicitly depend on the sparsity parameter of the projector. We investigate the performance of the approach for semidefinite relaxations appearing in polynomial optimization, with a focus on combinatorial optimization problems. In particular, we apply our method to the semidefinite relaxations of Maxcut and Max-2-sat . We show that for large unweighted graphs, we can obtain a good bound by solving a projection of the semidefinite relaxation of Maxcut . We also explore how to apply our method to find the stability number of four classes of imperfect graphs by solving a projection of the second level of the Lasserre Hierarchy. Overall, our computational experiments show that semidefinite programming problems appearing as relaxations of combinatorial optimization problems can be approximately solved using random projections as long as the number of constraints is not too large.

Application of an optimal class of antisymmetric wavelet filter banks for obstructive sleep apnea diagnosis using ECG signals

This work studies the problem of analyzing and, subsequently, optimizing the stabilization capabilities of a class of controllers for input constrained nonlinear systems. The proposed techniques apply to continuous, state feedback controllers which are defined in a subset of the state space where the time derivative of a known, candidate Control Lyapunov Function (CLF) can be made negative definite under the input constraints. This set is associated with the domain of this class of controllers and depends on the CLF. The analysis problem concerns approximating the domain and is posed via appropriately formulated set containment relationships through the generalized S-procedure with sum of squares (SOS) constraints. The optimization problem is concerned with the adjustment or enlargement of the domain and constitutes a way of controller synthesis. These objectives are pursued via optimizing over the coefficients of polynomial CLFs, through a sequence of semidefinite programming (SDP) problems with SOS constraints. By partitioning the state space based on the structure of the input value set and building upon earlier results on SOS methods, the SDP problems are subject to only convex constraints, rendering thus the proposed techniques computationally viable. The capabilities of the proposed algorithms are demonstrated through numerical examples.

This paper is concerned with obtaining the inverse of polynomial functions using semidefinite programming (SDP). Given a polynomial function and a nominal point at which the Jacobian of the function is invertible, the inverse function theorem states that the inverse of the polynomial function exists at a neighborhood of the nominal point. In this work, we show that this inverse function can be found locally using convex optimization. More precisely, we propose infinitely many SDPs, each of which finds the inverse function at a neighborhood of the nominal point. We also design a convex optimization to check the existence of an SDP problem that finds the inverse of the polynomial function at multiple nominal points and a neighborhood around each point. This makes it possible to identify an SDP problem (if any) that finds the inverse function over a large region. As an application, any system of polynomial equations can be solved by means of the proposed SDP problem whenever an approximate solution is available. The method developed in this work is numerically compared with Newton's method and the nuclear-norm technique.

Semidefinite programming (SDP) covers a wide range of applications such as robust optimization, polynomial optimization, combinatorial optimization, system and control theory, financial engineering, machine learning, quantum information and quantum chemistry. In those applications, SDP problems can be large scale easily. Such large scale SDP problems often satisfy a certain sparsity characterized by a chordal graph structure. This sparsity is classified in two types. The one is the domain space sparsity (d-space sparsity) for positive semidefinite symmetric matrix variables involved in SDP problems, and the other the range space sparsity (r-space sparsity) for matrix-inequality constraints in SDP problems. In this short note, we survey how we exploit these two types of sparsities to solve large scale linear and nonlinear SDP problems. We refer to the paper [7] for more details.