semidefinite programming (SDP) solvers are increasingly used as primitives in many program verification tasks to synthesize and verify polynomial invariants for a variety of systems including programs, hybrid systems ...
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semidefinite programming (SDP) solvers are increasingly used as primitives in many program verification tasks to synthesize and verify polynomial invariants for a variety of systems including programs, hybrid systems and stochastic models. On one hand, they provide a tractable alternative to reasoning about semi-algebraic constraints. However, the results are often unreliable due to numerical issues that include a large number of reasons such as floating-point errors, ill-conditioned problems, failure of strict feasibility, and more generally, the specifics of the algorithms used to solve SDPs. These issues influence whether the final numerical results are trustworthy or not. In this paper, we briefly survey the emerging use of SDP solvers in the static analysis community. We report on the perils of using SDP solvers for common invariant synthesis tasks, characterizing the common failures that can lead to unreliable answers. Next, we demonstrate existing tools for guaranteed semidefinite programming that often prove inadequate to our needs. Finally, we present a solution for verified semidefinite programming that can be used to check the reliability of the solution output by the solver and a padding procedure that can check the presence of a feasible nearby solution to the one output by the solver. We report on some successful preliminary experiments involving our padding procedure.
The paper describes a method for computing a lower bound of the global minimum of an indefinite quadratic form over a simplex. The bound is derived by computing an underestimator of the convex envelope by solving a se...
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The paper describes a method for computing a lower bound of the global minimum of an indefinite quadratic form over a simplex. The bound is derived by computing an underestimator of the convex envelope by solving a semidefinite program (SDP). This results in a convex quadratic program (QP). It is shown that the optimal value of the QP is a lower bound of the optimal value of the original problem. Since there exist fast (polynomial time) algorithms for solving SDP's and QP's the bound can be computed in reasonable time. Numerical experiments indicate that the relative error of the bound is about 10 percent for problems up to 20 variables, which is much better than a known SDP bound.
We study the asymptotic behavior of the interior-point bounds arising from the work of Yildirim and Todd on sensitivity analysis in semidefinite programming in comparison with the optimal partition bounds. We introduc...
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We study the asymptotic behavior of the interior-point bounds arising from the work of Yildirim and Todd on sensitivity analysis in semidefinite programming in comparison with the optimal partition bounds. We introduce a weaker notion of nondegeneracy and discuss its implications. For perturbations of the right-hand-side vector or the cost matrix, we show that the interior-point bounds evaluated on the central path using the Monteiro-Zhang family of search directions converge (as the duality gap tends to zero) to the symmetrized version of the optimal partition bounds under mild nondegeneracy assumptions. Furthermore, our analysis does not assume strict complementarity as long as the central path converges to the analytic center in a relatively controlled manner. We also show that the same convergence results carry over to iterates lying in an appropriate (very narrow) central path neighborhood if the Nesterov-Todd direction is used to evaluate the interior-point bounds. We extend our results to the case of simultaneous perturbations of the right-hand-side vector and the cost matrix. We also provide examples illustrating that our assumptions, in general, cannot be weakened.
The alternating-current optimal power flow (ACOPF) is one of the best known non-convex nonlinear optimization problems. We present a novel re-formulation of ACOPF, which is based on lifting the rectangular power-volta...
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The alternating-current optimal power flow (ACOPF) is one of the best known non-convex nonlinear optimization problems. We present a novel re-formulation of ACOPF, which is based on lifting the rectangular power-voltage rank-constrained formulation, and makes it possible to derive alternative semidefinite programming relaxations. For those, we develop a first-order method based on the parallel coordinate descent with a novel closed-form step based on roots of cubic polynomials.
We provide new tools for worst-case performance analysis of the gradient (or steepest descent) method of Cauchy for smooth strongly convex functions, and Newton's method for self-concordant functions, including th...
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We provide new tools for worst-case performance analysis of the gradient (or steepest descent) method of Cauchy for smooth strongly convex functions, and Newton's method for self-concordant functions, including the case of inexact search directions. The analysis uses semidefinite programming performance estimation, as pioneered by Drori and Teboulle [it Math. Program., 145 (2014), pp. 451-482], and extends recent performance estimation results for the method of Cauchy by the authors [it Optim. Lett., 11 (2017), pp. 1185-1199]. To illustrate the applicability of the tools, we demonstrate a novel complexity analysis of short step interior point methods using inexact search directions. As an example in this framework, we sketch how to give a rigorous worst-case complexity analysis of a recent interior point method by Abernethy and Hazan [it PMLR, 48 (2016), pp. 2520-2528].
Purpose This paper aims to propose a hand-eye calibration method of arc welding robot and laser vision sensor by using semidefinite programming (SDP). Design/methodology/approach The conversion relationship between th...
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Purpose This paper aims to propose a hand-eye calibration method of arc welding robot and laser vision sensor by using semidefinite programming (SDP). Design/methodology/approach The conversion relationship between the pixel coordinate system and laser plane coordinate system is established on the basis of the mathematical model of three-dimensional measurement of laser vision sensor. In addition, the conversion relationship between the arc welding robot coordinate system and the laser vision sensor measurement coordinate system is also established on the basis of the hand-eye calibration model. The ordinary least square (OLS) is used to calculate the rotation matrix, and the SDP is used to identify the direction vectors of the rotation matrix to ensure their orthogonality. Findings The feasibility identification can reduce the calibration error, and ensure the orthogonality of the calibration results. More accurate calibration results can be obtained by combining OLS + SDP. Originality/value A set of advanced calibration methods is systematically established, which includes parameters calibration of laser vision sensor and hand-eye calibration of robots and sensors. For the hand-eye calibration, the physics feasibility problem of rotating matrix is creatively put forward, and is solved through SDP algorithm. High-precision calibration results provide a good foundation for future research on seam tracking.
We present algorithms for the soluition of two problems in array pattern synthesis. The first is the design of nonuniform arrays with a desired magnitude response. The second is that of robust design, i.e., design in ...
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We present algorithms for the soluition of two problems in array pattern synthesis. The first is the design of nonuniform arrays with a desired magnitude response. The second is that of robust design, i.e., design in the presence of uncertainties. Constraints such as power limitation can be addressed with both problems. The algorithms that we present are based on semidefinite programming, for which efficient software is readily available. We present examples that illustrate the effectiveness of our approach.
We discuss computational enhancements for the low-rank semidefinite programming algorithm, including the extension to block semidefinite programs (SDPs), an exact linesearch procedure, and a dynamic rank reduction sch...
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We discuss computational enhancements for the low-rank semidefinite programming algorithm, including the extension to block semidefinite programs (SDPs), an exact linesearch procedure, and a dynamic rank reduction scheme. A truncated-Newton method is also introduced, and several preconditioning strategies are proposed. Numerical experiments illustrating these enhancements are provided on a wide class of test problems. In particular, the truncated-Newton variant is able to achieve high accuracy in modest amounts of time on maximum-cut-type SDPs.
This paper deals with nonlinear smooth optimization problems with equality and inequality constraints, as well as semidefinite constraints on nonlinear symmetric matrix-valued functions. A new semidefinite programming...
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This paper deals with nonlinear smooth optimization problems with equality and inequality constraints, as well as semidefinite constraints on nonlinear symmetric matrix-valued functions. A new semidefinite programming algorithm that takes advantage of the structure of the matrix constraints is presented. This one is relevant in applications where the matrices have a favorable structure, as in the case when finite element models are employed. FDIPA_GSDP is then obtained by integration of this new method with the well known Feasible Direction Interior Point Algorithm for nonlinear smooth optimization, FDIPA. FDIPA_GSDP makes iterations in the primal and dual variables to solve the first order optimality conditions. Given an initial feasible point with respect to the inequality constraints, FDIPA_GSDP generates a feasible descent sequence, converging to a local solution of the problem. At each iteration a feasible descent direction is computed by merely solving two linear systems with the same matrix. A line search along this direction looks for a new feasible point with a lower objective. Global convergence to stationary points is proved. Some structural optimization test problems were solved very efficiently, without need of parameters tuning.
Many theoretical and algorithmic results in semidefinite programming are based on the assumption that Slater's constraint qualification is satisfied for the primal and the associated dual problem. We consider semi...
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Many theoretical and algorithmic results in semidefinite programming are based on the assumption that Slater's constraint qualification is satisfied for the primal and the associated dual problem. We consider semidefinite problems with zero duality gap for which Slater's condition fails for at least one of the primal and dual problem. We propose a numerically reasonable way of dealing with such semidefinite programs. The new method is based on a standard search direction with damped Newton steps towards primal and dual feasibility.
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