# Numerical Results

In this section, we present statistics for the numerical results obtained by VSDP for conic programming problems. The tests were performed using approximations computed by the conic solvers: CSDP, SEDUMI, SDPT3, SDPA, LINPROG, and LPSOLVE. For second order cone programming problems only SEDUMI and SDPT3 were used. The solvers have been called with their default parameters. Almost all of the problems that could not be solved with a guaranteed accuracy about $10^{-7}$ are known to be ill-posed (cf. [Freund2003]).

We measure the difference between two numbers by the frequently used quantity $% \label{eq:accurracy_measure} ??(a,b) := \dfrac{a-b}{\max\{1.0, (a+b)/2\}}. %]]>$

Notice that we do not use the absolute value of $a - b$. Hence, a negative sign implies that $a < b$.

## SDPLIB

In the following, we describe the numerical results for problems from the SDPLIB suite of Borchers [Borchers1999]. In [Freund2007] it is shown that four problems are infeasible, and 32 problems are ill-posed.

VSDP could compute rigorous bounds of the optimal values for all feasible well-posed problems and verify the existence of strictly primal and dual feasible solutions. Hence, strong duality is proved. For the 32 ill-posed problems VSDP has computed the upper bound $fU = \infty$, which reflects the fact that the distance to the next primal infeasible problem is zero. For the four infeasible problems VSDP could compute rigorous certificates of infeasibility. Detailed numerical results can be found in Table benchmark_sdplib_2012_12_12.html, where the computed rigorous upper bound $fU$, the rigorous lower bound $fL$, and the rigorous error bound $??(fU,fL)$ are displayed. We have set $??(fU,fL) = NaN$ if the upper or the lower bound is infinite. Table benchmark_sdplib_2012_12_12.html also contains running times in seconds, where $t_{s}$ is the time for computing the approximations, and $t_{u}$ and $t_{l}$ are the times for computing the upper and the lower rigorous error bounds, respectively.

Some major characteristics of our numerical results for the SDPLIB are summarized below.

It displays in the first row the median $\operatorname{med}(??(fU,fL))$ of the computed error bounds, in the second row the largest error bound $\max(??(fU,fL))$, and in the third row the minimal error bound $\min(??(fU,fL))$. For this statistic only the well-posed problems are taken into account. In the two last rows the medians of time ratios $t_{u} / t_{s}$ and $t_{l} / t_{s}$ are displayed.

The median of $??(fU,fL)$ shows that for all conic solvers rigorous error bounds with 7 or 8 significant decimal digits could be computed for most problems.

Furthermore, the table shows that the error bounds as well as the time ratios depend significantly on the used conic solver. In particular the resulting time ratios indicate that the conic solvers CSDP and SeDuMi aim to compute approximate primal interior $??$-optimal solutions. In contrast SDPA and SDPT3 aim to compute dual interior $??$-optimal solutions.

Even the largest problem MaxG60 with about 24 million variables and 7000 constraints can be solved rigorously by VSDP, with high accuracy and in a reasonable time.

## NETLIB LP

Here we describe some numerical results for the NETLIB linear programming library. This is a well known test suite containing many difficult to solve, real-world examples from a variety of sources.

For this test set Ord????ez and Freund [Freund2003] have shown that 71 % of the problems are ill-posed. This statement is well reflected by our results: for the ill-posed problems VSDP computed infinite lower or infinite upper bounds. This happens if the distance to the next dual infeasible or primal infeasible problem is zero, respectively.

For the computation of approximations we used the solvers LINPROG, LPSOLVE, SEDUMI, and SDPT3. In the following table we display the same quantities as in the previous section. Again only the well-posed problems are taken into account.

Here we would like to mention also the numerical results of the C++ software package LURUPA [Keil2006], [Keil2009]. In [Keil2008] comparisons with other software packages for the computation of rigorous errors bounds are described.

Detailed results can be found in Table benchmark_netlib_lp_2012_12_12.html.

## DIMACS

We present some statistics of numerical results for the DIMACS test library of semidefinte-quadratic-linear programs. This library was assembled for the purposes of the 7-th DIMACS Implementation Challenge. There are about 50 challenging problems that are divided into 12 groups. For details see [Pataki2002]. In each group there are about 5 instances, from routinely solvable ones reaching to those at or beyond the capabilities of current solvers. Due to limited available memory some problems of the DIMACS test library have been omitted in our test. These are: coppo68, industry2, fap09, fap25, fap36, hamming112, and qssp180old.

One of the largest problems which could be solved by VSDP is the problem hamming102, with 23041 equality constraints and 1048576 variables.

The approximate solvers SDPA and CSDP are designed for SDP problems only. Hence, in the table above we have displayed for these solvers only the statistic of the SDP problems.

Detailed results can be found in Table benchmark_dimacs_2012_12_12.html.

## Kovcara’s Library of Structural Optimization Problems

In this section a statistic of the numerical results for problems from structural and topological optimization is presented. Structural and especially free material optimization gains more and more interest in the recent years. The most prominent example is the design of ribs in the leading edge of the new Airbus A380. We performed tests on problems from the test library collected by Kocvara. This is a collection of 26 sparse semidefinite programming problems. More details on these problems can be found in [BenTal2000], [Kocvara2002], and [Bendsoe1997]. For 24 problems out of this collection VSDP could compute a rigorous primal and dual $??$-optimal solution. Caused by the limited available memory and the great computational times the largest problems mater-5, mater-6, shmup5, trto5, and vibra5 has been tested only with the solver SDPT3. The largest problem that was rigorously solved by VSDP is shmup5. This problem has 1800 equality constraints and 13849441 variables.

A statistic of these numerical experiments is given below:

Detailed results can be found in Table benchmark_kovcara_2012_12_12.html.

Published with GNU Octave 4.4.1